Problem 57983. Easy Sequences 109: Summation of Derivatives of a Trigonometric Function

Created by Ramon Villamangca

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>A trigonometric function, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\text{T}(x)</w:t></w:r></w:customXml><w:r><w:t>,  is defined as follows:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>\\text{T}(x)=\\frac^{1+2\\tan(x)-\\tan^2(x)}_{1+\\tan^2(x)</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                where:     </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\tan^2(x) = \\tan(x)\\cdot\\tan(x)</w:t></w:r></w:customXml><w:r><w:t>; and </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:t> is in radians.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>In this problem we are asked to evaluate the following summation:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>s=\\text{S}(x,n)=\\sum_{k=0}^n \\frac^{\\text{T}^{(k)}(x)}_{2^k}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                where:    </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:t> is a real number; </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> is an integer; </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\text{T}^{(k)}(x)=\\frac{\\mathrm{d}^k}{\\mathrm{d}x^k}\\text{T}(x)</w:t></w:r></w:customXml><w:r><w:t> or the </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k</w:t></w:r></w:customXml><w:r><w:t>-th derivative of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\text{T}</w:t></w:r></w:customXml><w:r><w:t> with respect to </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:t>; and </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\text{T}^{(0)}(x)=\\text{T}(x)</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For example for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x=2</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n=2</w:t></w:r></w:customXml><w:r><w:t>, we have:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[  >>  syms x;\n  >>  T = (1+2*tan(x)-tan(x)^2) / (1+tan(x)^2);\n  >>  S = T/2^0 + diff(T)/2^1 + diff(T,2)/2^2;\n  >>  s = vpa(subs(S,x,2))\n        s = 0.10315887444431633673347091141408;]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Please present the final output rounded-off to 6 decimal places. Therefore the final answer is </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>s=0.103159</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>-------------------------</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>NOTE: </w:t></w:r><w:r><w:t>Symbolic toolbox is not available to Cody players. It is possible to do </w:t></w:r><w:hyperlink w:docLocation=\"https://www.unioviedo.es/compnum/labs/lab07_der_int/lab07_der_int.html\"><w:r><w:rPr><w:b/></w:rPr><w:t>numerical differentiation</w:t></w:r></w:hyperlink><w:r><w:t> in Matlab without symbolic toolbox.  Just make sure that the output would be accurate within 6 decimal places of the 'exact' value obtained using </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>syms</w:t></w:r><w:r><w:t>, as shown above.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; "><div style="block-size: 622px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 311px; transform-origin: 407px 311px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">A trigonometric function, </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="31.5" height="19" style="width: 31.5px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">,  is defined as follows:</span></span></div><div style="block-size: 38px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19px; text-align: left; transform-origin: 384px 19px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                </span></span><span style="vertical-align:-14px"><img src="data:image/png;base64,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" width="133.5" height="38" style="width: 133.5px; height: 38px;"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                where:     </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="152.5" height="20" style="width: 152.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">; and </span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                                </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">x</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> is in radians.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">In this problem we are asked to evaluate the following summation:</span></span></div><div style="block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                </span></span><span style="vertical-align:-17px"><img src="data:image/png;base64,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" width="142.5" height="46" style="width: 142.5px; height: 46px;"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                where:    </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">x</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> is a real number; </span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                                </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> is an integer; </span></span></div><div style="block-size: 37px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.5px; text-align: left; transform-origin: 384px 18.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                                </span></span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="112.5" height="37" style="width: 112.5px; height: 37px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> or the </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">k</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">-th derivative of </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);">T</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> with respect to </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">x</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">; and </span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                                </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="91" height="20" style="width: 91px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">For example for </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, we have:</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 100px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 50px; transform-origin: 404px 50px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">  &gt;&gt;  syms x;</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">  &gt;&gt;  T = (1+2*tan(x)-tan(x)^2) / (1+tan(x)^2);</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">  &gt;&gt;  S = T/2^0 + diff(T)/2^1 + diff(T,2)/2^2;</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">  &gt;&gt;  s = vpa(subs(S,x,2))</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">        s = 0.10315887444431633673347091141408;</span></span></div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Please present the final output rounded-off to 6 decimal places. Therefore the final answer is </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="83.5" height="18" style="width: 83.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">-------------------------</span></span></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">NOTE: </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Symbolic toolbox is not available to Cody players. It is possible to do </span></span><a target='_blank' href = "https://www.unioviedo.es/compnum/labs/lab07_der_int/lab07_der_int.html"><span style=""><span style="font-weight: 700; ">numerical differentiation</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> in Matlab without symbolic toolbox.  Just make sure that the output would be accurate within 6 decimal places of the 'exact' value obtained using </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-weight: 700; ">syms</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, as shown above.</span></span></div></div></div>

A trigonometric function, ,  is defined as follows:
                
                where:     ; and 
                                 is in radians.
In this problem we are asked to evaluate the following summation:
                
                where:     is a real number; 
                                 is an integer; 
                                 or the -th derivative of  with respect to ; and 
                                .
For example for  and , we have:
  >>  syms x;
  >>  T = (1+2*tan(x)-tan(x)^2) / (1+tan(x)^2);
  >>  S = T/2^0 + diff(T)/2^1 + diff(T,2)/2^2;
  >>  s = vpa(subs(S,x,2))
        s = 0.10315887444431633673347091141408;
Please present the final output rounded-off to 6 decimal places. Therefore the final answer is .
-------------------------
NOTE: Symbolic toolbox is not available to Cody players. It is possible to do numerical differentiation in Matlab without symbolic toolbox.  Just make sure that the output would be accurate within 6 decimal places of the 'exact' value obtained using syms, as shown above.

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Problem 57983. Easy Sequences 109: Summation of Derivatives of a Trigonometric Function

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