{"id":"51249a16-a92c-4898-b1a4-9787687b554e","title":"Maclaurin Series Expansion for Tan(x)","created_at":"2025-06-16T19:04:27Z","saved":true,"shared":true,"branch_id":"00000000-0000-4000-0000-000000000000","profile":{"id":null,"name":null,"model":"gemini-2-5-flash","model_name":"Gemini 2.5 Flash","model_provider":"google","model_input_limit":1000000,"recommended":false,"model_info":"\n\n<div class=\"heading\">\n <div class=\"left\">\n \n \n \n <svg class=\"model-icon icon-xs\" data-model=\"gemini-2-5-flash\">\n <use href=\"#icon-google\"/>\n </svg>\n <h3>\n \n <span class=\"model-provider\">Google</span>\n \n <span class=\"model-name\">none</span>\n </h3>\n </div>\n\n \n <a href=\"/settings?p=custom_assistant&id=none\">\n <svg class=\"icon-sm edit-icon\" aria-hidden=\"true\">\n <use href=\"#icon-pencil\" />\n </svg>\n </a>\n \n</div>\n\n\n<hr />\n<ul>\n \n <li>\n <h4>\n Cost\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Relatively cheap\">\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --low\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --low\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-cost --low\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-cost --low\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-cost --low\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Quality\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 2 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Speed\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 5 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Privacy\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 4 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li class=\"divider\" aria-hidden=\"true\"></li>\n <li>\n <h4>Last updated</h4>\n <span class=\"date\">\n \n \n 2025-06-17\n \n \n </span>\n </li>\n \n</ul>\n","model_provider_name":"Google","internet_access":true,"personalizations":true,"shortcut":null,"is_default_profile":false},"tag_ids":[],"read_only":true}

[{"id":"9823b854-8040-4978-9e74-8223fa7ffeed","thread_id":"51249a16-a92c-4898-b1a4-9787687b554e","created_at":"2025-06-16T19:04:27Z","state":"done","prompt":"Show me the Maclaurin series expansion for tan(x).","references_html":"<ol data-ref-list>\n \n <li>\n <a href=\"https://en.wikipedia.org/wiki/Taylor_series\" >Taylor series - Wikipedia</a>\n <span class=\"__domain-name\">en.wikipedia.org</span>\n <div class=\"trigger\" tabindex=\"0\">\n <span class=\"contribution\" style=\"background: linear-gradient(to right, var(--primary-70) , transparent )\"></span>\n <div class=\"tooltip top end tooltip_base tw-auto\">\n This percentage indicates how much a source has contributed to the answer.\n </div>\n </div>\n </li>\n \n <li>\n <a href=\"https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx.html\" >Maclaurin Series tan x - Peter Vis</a>\n <span class=\"__domain-name\">www.petervis.com</span>\n <div class=\"trigger\" tabindex=\"0\">\n <span class=\"contribution\" style=\"background: linear-gradient(to right, var(--primary-70) , transparent )\"></span>\n <div class=\"tooltip top end tooltip_base tw-auto\">\n This percentage indicates how much a source has contributed to the answer.\n </div>\n </div>\n </li>\n \n <li>\n <a href=\"https://math.stackexchange.com/questions/1577978/maclaurins-series-for-secx-with-help-of-maclaurins-series-for-tanx\" >calculus - Maclaurin&#39;s Series for $\\sec(x)$ with help of...</a>\n <span class=\"__domain-name\">math.stackexchange.com</span>\n <div class=\"trigger\" tabindex=\"0\">\n <span class=\"contribution\" style=\"background: linear-gradient(to right, var(--primary-70) , transparent )\"></span>\n <div class=\"tooltip top end tooltip_base tw-auto\">\n This percentage indicates how much a source has contributed to the answer.\n </div>\n </div>\n </li>\n \n <li>\n <a href=\"https://math.stackexchange.com/questions/1546539/maclaurin-polynomial-of-tanx\" >Maclaurin polynomial of tan(x) - Math Stack Exchange</a>\n <span class=\"__domain-name\">math.stackexchange.com</span>\n <div class=\"trigger\" tabindex=\"0\">\n <span class=\"contribution\" style=\"background: linear-gradient(to right, var(--primary-70) , transparent )\"></span>\n <div class=\"tooltip top end tooltip_base tw-auto\">\n This percentage indicates how much a source has contributed to the answer.\n </div>\n </div>\n </li>\n \n</ol>\n","references_md":"[^1]: [Taylor series - Wikipedia](https://en.wikipedia.org/wiki/Taylor_series)\n[^2]: [Maclaurin Series tan x - Peter Vis](https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx.html)\n[^3]: [calculus - Maclaurin's Series for $\\sec(x)$ with help of...](https://math.stackexchange.com/questions/1577978/maclaurins-series-for-secx-with-help-of-maclaurins-series-for-tanx)\n[^4]: [Maclaurin polynomial of tan(x) - Math Stack Exchange](https://math.stackexchange.com/questions/1546539/maclaurin-polynomial-of-tanx)\n","reply":"<details><summary>Searching with Kagi:<p></p>\n<ul>\n<li>Maclaurin series expansion for tan(x)</li></ul></summary> Sources (10):<br>\n<strong>Taylor series - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Taylor_series\" target=\"_blank\">https://en.wikipedia.org/wiki/Taylor_series</a><br>\nMay 18, 2025 1 With a = 0 , the Maclaurin series takes the form: 2 f ( 0 ) + f ( 0 ) 1 ! x + f ( 0 ) 2 ! x 2 + = n = 0 f ( n ) ( 0 ) n ! x n .\n\n<p><strong>Maclaurin Series tan x - Peter Vis</strong><br>\n<a href=\"https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx.html\" target=\"_blank\">https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx.html</a><br>\nNov 03, 2014 Deriving the Maclaurin series for tan x is a very simple process. It is more of an exercise in differentiating using the chain rule to find the derivatives.</p>\n<p><strong>Maclaurin Series</strong><br>\n<a href=\"https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series.html\" target=\"_blank\">https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series.html</a><br>\nSep 17, 2017 The expansion above shows the general formula of the Maclaurin series. I have animated it to make it simpler for students to understand. The series is one of the easiest to understand in mathematics and very repetitive as you can imagine.</p>\n<p><strong>Maclaurin Series xe^x</strong><br>\n<a href=\"https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_xex.html\" target=\"_blank\">https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_xex.html</a><br>\nOct 15, 2017 expansion. The Maclaurin series expansion for xe^x is very easy to derive. This is one of the easiest ones to do because the derivatives are very easy to find.</p>\n<p><strong>Maclaurin polynomial of tan(x) - Math Stack Exchange</strong><br>\n<a href=\"https://math.stackexchange.com/questions/1546539/maclaurin-polynomial-of-tanx\" target=\"_blank\">https://math.stackexchange.com/questions/1546539/maclaurin-polynomial-of-tanx</a><br>\nNov 25, 2015 tanx=∞∑n=0A2n+1⋅x2n+1(2n+1)! , where A2n+1 are the so-called “zag-” or tangent numbers. They go hand-in-hand with their complementary ...</p>\n<p><strong>taylor expansion - Expand <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>tan</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>x</mi></mrow></math></span> as infinite</strong><br>\n<a href=\"https://math.stackexchange.com/questions/2291487/expand-tan-1x-as-infinite-series\" target=\"_blank\">https://math.stackexchange.com/questions/2291487/expand-tan-1x-as-infinite-series</a><br>\nMar 28, 2025 How can we obtain the infinite series for <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>tan</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>?</mo></mrow></math></span> Finding the derivatives in Taylor series becomes difficult.</p>\n<p><strong>Evaluate $\\sum</strong><br>\n<a href=\"https://math.stackexchange.com/questions/4954096/evaluate-sum-n-1-infty-frac-left-1-rightn-left2n1-right-t\" target=\"_blank\">https://math.stackexchange.com/questions/4954096/evaluate-sum-n-1-infty-frac-left-1-rightn-left2n1-right-t</a><br>\nMar 31, 2025 ... Maclaurin series for sine: $$ \\sin(x) = \\sum_{n=0}^\\infty ... I have found a Taylor series and a Maclaurin series for a function about x=0....</p>\n<p><strong>trigonometric series - What is the advantage of using maclaurin...</strong><br>\n<a href=\"https://math.stackexchange.com/questions/2654124/what-is-the-advantage-of-using-maclaurin-expansion-of-functions-like-tanx-over\" target=\"_blank\">https://math.stackexchange.com/questions/2654124/what-is-the-advantage-of-using-maclaurin-expansion-of-functions-like-tanx-over</a><br>\nFeb 17, 2018 Problem with Maclaurin series expansion method.Using the Maclaurin series when calculating limits.</p>\n<p><strong>taylor expansion - Expand <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msqrt><mrow><mn>1</mn><mo>−</mo><mn>4</mn><mi>x</mi></mrow></msqrt></mrow></math></span> into an</strong><br>\n<a href=\"https://math.stackexchange.com/questions/4940241/expand-sqrt1-4x-into-an-infinite-power-series\" target=\"_blank\">https://math.stackexchange.com/questions/4940241/expand-sqrt1-4x-into-an-infinite-power-series</a><br>\nFeb 26, 2025 The rest boils down to finding a reasonable bound on the size of <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>f</mi><mrow><mo stretchy=\"false\">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> for values of <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi></mrow></math></span> bounded away from the endpoints of the interval ...</p>\n<p><strong>calculus - Maclaurin's Series for <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>sec</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> with help of...</strong><br>\n<a href=\"https://math.stackexchange.com/questions/1577978/maclaurins-series-for-secx-with-help-of-maclaurins-series-for-tanx\" target=\"_blank\">https://math.stackexchange.com/questions/1577978/maclaurins-series-for-secx-with-help-of-maclaurins-series-for-tanx</a><br>\nApr 25, 2017 with the help of Maclaurin's series for tan(x).It is probably not the simplest way to do it but, using ddxtan(x)=sec2(x). you will need to compute tan(x)=x+x33+2x515+17x7315+O(x9).</p></details><p>The Maclaurin series is a special case of the Taylor series expansion of a function about 0. The general formula for a Maclaurin series is: </p>\n<p>$f(x) = \\sum_{n=0}^{\\infty} \\frac{f^{(n)}(0)}{n!} x^n = f(0) + f'(0)x + \\frac{f''(0)}{2!}x^2 + \\frac{f'''(0)}{3!}x^3 + \\dots$<sup class=\"trigger\"><a href=\"https://en.wikipedia.org/wiki/Taylor_series\">1</a><span class=\"tooltip top center citation\"><a href=\"https://en.wikipedia.org/wiki/Taylor_series#:~:text=1%20With,n%20.\"><span class=\"source\"><img alt=\"\" src=\"/proxy/favicons?c=lQ30-cxV_E2Vdyqb6hrbuYgId7qu7hecjO7o1T7aqQDFc7XpcLttL1Ryala-GNHJuWyqBKj8MT4ZBZYnSH4uzuNLNQdbiyz2-gUDYI6hGs0%3D\"> en.wikipedia.org</span><b>1. Taylor series - Wikipedia</b></a><span class=\"passage\">1 With a = 0 , the Maclaurin series takes the form: 2 f ( 0 ) + f ( 0 ) 1 ! x + f ( 0 ) 2 ! x 2 + = n = 0 f ( n ) ( 0 ) n ! x n .</span></span></sup></p>\n<p>To find the Maclaurin series for <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span>, we need to find the derivatives of <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> evaluated at <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span>:<sup class=\"trigger\"><a href=\"https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx.html\">2</a><span class=\"tooltip top center citation\"><a href=\"https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx.html#:~:text=Deriving%20the,the%20derivatives.\"><span class=\"source\"><img alt=\"\" src=\"/proxy/favicons?c=lQ30-cxV_E2Vdyqb6hrbuYgId7qu7hecjO7o1T7aqQDFc7XpcLttL1Ryala-GNHJcb7XdpHQ6JSttkxyCDJpSPDtn8iNlZ1qAkpfZbgz7IU%3D\"> www.petervis.com</span><b>2. Maclaurin Series tan x - Peter Vis</b></a><span class=\"passage\">Deriving the Maclaurin series for tan x is a very simple process. It is more of an exercise in differentiating using the chain rule to find the derivatives.</span></span></sup></p>\n<ul>\n<li><span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>⇒</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mi>tan</mi><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mn>0</mn></mrow></math></span></li>\n<li><span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>f</mi><mi>′</mi></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msup><mi>sec</mi><mn>2</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>⇒</mo><msup><mi>f</mi><mi>′</mi></msup><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>=</mo><msup><mi>sec</mi><mn>2</mn></msup><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn></mrow></math></span></li>\n<li><span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>f</mi><mi>″</mi></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>2</mn><mi>sec</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mi>·</mi><mo stretchy=\"false\">(</mo><mi>sec</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>=</mo><mn>2</mn><msup><mi>sec</mi><mn>2</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>⇒</mo><msup><mi>f</mi><mi>″</mi></msup><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mn>2</mn><msup><mi>sec</mi><mn>2</mn></msup><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mi>tan</mi><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mn>0</mn></mrow></math></span></li>\n<li>$f'''(x) = 2(2\\sec(x)(\\sec(x)\\tan(x))\\tan(x) + \\sec^2(x)\\sec^2(x)) = 4\\sec^2(x)\\tan^2(x) + 2\\sec^4(x) \\Rightarrow f'''(0) = 4\\sec^2(0)\\tan^2(0) + 2\\sec^4(0) = 2$</li>\n<li><span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>f</mi><mrow><mo stretchy=\"false\">(</mo><mn>4</mn><mo stretchy=\"false\">)</mo></mrow></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>8</mn><msup><mi>sec</mi><mn>2</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><msup><mi>tan</mi><mn>3</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>+</mo><mn>8</mn><msup><mi>sec</mi><mn>4</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>+</mo><mn>8</mn><msup><mi>sec</mi><mn>3</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">(</mo><mi>sec</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>=</mo><mn>8</mn><msup><mi>sec</mi><mn>2</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><msup><mi>tan</mi><mn>3</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>+</mo><mn>16</mn><msup><mi>sec</mi><mn>4</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>⇒</mo><msup><mi>f</mi><mrow><mo stretchy=\"false\">(</mo><mn>4</mn><mo stretchy=\"false\">)</mo></mrow></msup><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mn>0</mn></mrow></math></span></li>\n<li><span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>f</mi><mrow><mo stretchy=\"false\">(</mo><mn>5</mn><mo stretchy=\"false\">)</mo></mrow></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>8</mn><mo stretchy=\"false\">(</mo><mn>2</mn><msup><mi>sec</mi><mn>2</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><msup><mi>tan</mi><mn>4</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>+</mo><mn>3</mn><msup><mi>sec</mi><mn>4</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><msup><mi>tan</mi><mn>2</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>+</mo><mn>16</mn><mo stretchy=\"false\">(</mo><mn>4</mn><msup><mi>sec</mi><mn>4</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><msup><mi>tan</mi><mn>2</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>+</mo><msup><mi>sec</mi><mn>6</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>⇒</mo><msup><mi>f</mi><mrow><mo stretchy=\"false\">(</mo><mn>5</mn><mo stretchy=\"false\">)</mo></mrow></msup><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mn>16</mn><msup><mi>sec</mi><mn>6</mn></msup><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mn>16</mn></mrow></math></span></li>\n</ul>\n<p>Substituting these values into the Maclaurin series formula:</p>\n<p><span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>0</mn><mo>+</mo><mn>1</mn><mi>x</mi><mo>+</mo><mfrac><mrow><mn>0</mn></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mrow><mn>0</mn></mrow><mrow><mn>4</mn><mo>!</mo></mrow></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mfrac><mrow><mn>16</mn></mrow><mrow><mn>5</mn><mo>!</mo></mrow></mfrac><msup><mi>x</mi><mn>5</mn></msup><mo>+</mo><mo>…</mo></mrow></math></span><br>\n<span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>6</mn></mrow></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mrow><mn>16</mn></mrow><mrow><mn>120</mn></mrow></mfrac><msup><mi>x</mi><mn>5</mn></msup><mo>+</mo><mo>…</mo></mrow></math></span><br>\n<span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>15</mn></mrow></mfrac><msup><mi>x</mi><mn>5</mn></msup><mo>+</mo><mo>…</mo></mrow></math></span><sup class=\"trigger\"><a href=\"https://math.stackexchange.com/questions/1577978/maclaurins-series-for-secx-with-help-of-maclaurins-series-for-tanx\">3</a><span class=\"tooltip top center citation\"><a href=\"https://math.stackexchange.com/questions/1577978/maclaurins-series-for-secx-with-help-of-maclaurins-series-for-tanx#:~:text=with%20the,compute%20tan%28x%29%3Dx%2Bx33%2B2x515%2B17x7315%2BO%28x9%29.\"><span class=\"source\"><img alt=\"\" src=\"/proxy/favicons?c=lQ30-cxV_E2Vdyqb6hrbuYgId7qu7hecjO7o1T7aqQDFc7XpcLttL1Ryala-GNHJFHTCkIAAMVKWEtLPMaZTucjhcvHgwKfPEdICIoJwkrL0UpLDHwoGN6lfUT54uPp2\"> math.stackexchange.com</span><b>3. calculus - Maclaurin's Series for $\\sec(x)$ with help of...</b></a><span class=\"passage\">with the help of Maclaurin's series for tan(x).It is probably not the simplest way to do it but, using ddxtan(x)=sec2(x). you will need to compute tan(x)=x+x33+2x515+17x7315+O(x9).</span></span></sup></p>\n<p>The Maclaurin series for <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> can be expressed as:<br>\n<span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>tan</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mi>A</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mfrac><mrow><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><mo stretchy=\"false\">(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy=\"false\">)</mo><mo>!</mo></mrow></mfrac></mrow></math></span>, where <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>A</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></math></span> are the tangent numbers.<sup class=\"trigger\"><a href=\"https://math.stackexchange.com/questions/1546539/maclaurin-polynomial-of-tanx\">4</a><span class=\"tooltip top center citation\"><a href=\"https://math.stackexchange.com/questions/1546539/maclaurin-polynomial-of-tanx#:~:text=tanx%3D%E2%88%9E%E2%88%91n%3D0A2n%2B1%E2%8B%85x2n%2B1%282n%2B1%29%21%20%2C,their%20complementary\"><span class=\"source\"><img alt=\"\" src=\"/proxy/favicons?c=lQ30-cxV_E2Vdyqb6hrbuYgId7qu7hecjO7o1T7aqQDFc7XpcLttL1Ryala-GNHJFHTCkIAAMVKWEtLPMaZTucjhcvHgwKfPEdICIoJwkrL0UpLDHwoGN6lfUT54uPp2\"> math.stackexchange.com</span><b>4. Maclaurin polynomial of tan(x) - Math Stack Exchange</b></a><span class=\"passage\">tanx=∞∑n=0A2n+1⋅x2n+1(2n+1)! , where A2n+1 are the so-called “zag-” or tangent numbers. They go hand-in-hand with their complementary ...</span></span></sup></p>","md":"The Maclaurin series is a special case of the Taylor series expansion of a function about 0. The general formula for a Maclaurin series is: \n\n$f(x) = \\sum_{n=0}^{\\infty} \\frac{f^{(n)}(0)}{n!} x^n = f(0) + f'(0)x + \\frac{f''(0)}{2!}x^2 + \\frac{f'''(0)}{3!}x^3 + \\dots$[^1]\n\nTo find the Maclaurin series for $\\tan(x)$, we need to find the derivatives of $\\tan(x)$ evaluated at $x=0$:[^2]\n\n* $f(x) = \\tan(x) \\Rightarrow f(0) = \\tan(0) = 0$\n* $f'(x) = \\sec^2(x) \\Rightarrow f'(0) = \\sec^2(0) = 1$\n* $f''(x) = 2\\sec(x) \\cdot (\\sec(x)\\tan(x)) = 2\\sec^2(x)\\tan(x) \\Rightarrow f''(0) = 2\\sec^2(0)\\tan(0) = 0$\n* $f'''(x) = 2(2\\sec(x)(\\sec(x)\\tan(x))\\tan(x) + \\sec^2(x)\\sec^2(x)) = 4\\sec^2(x)\\tan^2(x) + 2\\sec^4(x) \\Rightarrow f'''(0) = 4\\sec^2(0)\\tan^2(0) + 2\\sec^4(0) = 2$\n* $f^{(4)}(x) = 8\\sec^2(x)\\tan^3(x) + 8\\sec^4(x)\\tan(x) + 8\\sec^3(x)(\\sec(x)\\tan(x)) = 8\\sec^2(x)\\tan^3(x) + 16\\sec^4(x)\\tan(x) \\Rightarrow f^{(4)}(0) = 0$\n* $f^{(5)}(x) = 8(2\\sec^2(x)\\tan^4(x) + 3\\sec^4(x)\\tan^2(x)) + 16(4\\sec^4(x)\\tan^2(x) + \\sec^6(x)) \\Rightarrow f^{(5)}(0) = 16\\sec^6(0) = 16$\n\nSubstituting these values into the Maclaurin series formula:\n\n$\\tan(x) = 0 + 1x + \\frac{0}{2!}x^2 + \\frac{2}{3!}x^3 + \\frac{0}{4!}x^4 + \\frac{16}{5!}x^5 + \\dots$\n$\\tan(x) = x + \\frac{2}{6}x^3 + \\frac{16}{120}x^5 + \\dots$\n$\\tan(x) = x + \\frac{1}{3}x^3 + \\frac{2}{15}x^5 + \\dots$[^3]\n\nThe Maclaurin series for $\\tan(x)$ can be expressed as:\n$\\tan(x) = \\sum_{n=0}^{\\infty} A_{2n+1} \\frac{x^{2n+1}}{(2n+1)!}$, where $A_{2n+1}$ are the tangent numbers.[^4]","profile":{"id":null,"name":null,"model":"gemini-2-5-flash","model_name":"Gemini 2.5 Flash","model_provider":"google","model_input_limit":1000000,"recommended":false,"model_info":"\n\n<div class=\"heading\">\n <div class=\"left\">\n \n \n \n <svg class=\"model-icon icon-xs\" data-model=\"gemini-2-5-flash\">\n <use href=\"#icon-google\"/>\n </svg>\n <h3>\n \n <span class=\"model-provider\">Google</span>\n \n <span class=\"model-name\">none</span>\n </h3>\n </div>\n\n \n 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