Questions regarding the Taylor series expansion of univariate and multivariate functions, including coefficients and bounds on remainders. A special case is also known as the Maclaurin series.

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Understanding a taylor expansion

If $$s(y) = \begin{cases} 2\sin(y/2)/y & \text{if $y \neq 0$} \\ 1 & \text{otherwise} \end{cases}$$ why is the taylor expansion of $g(y) = \frac{1}{s(y)}$: $$ \frac{1}{s(y)} = 1 + ...
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50 views

Bounding the remainder

I need to find the 3rd order Taylor polynomials and bound the remainder term at $(0,0)$. The function is $$f(x,y)=\cos(x)\sin(y)$$ Here is what I did: first, I found the taylor expansions of sin and ...
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An Expression of $\ln(2)$

I saw online that the following infinite series has a value of $\ln(2)$: $\sum_{n=0}^\infty \left(\dfrac{1}{n+1}-\dfrac{1}{n+2} +\dfrac{1}{n+3}-\cdot\cdot\cdot\right)^2$ I started off by defining ...
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$\ln (2)$ from Taylor series with error at most $10^{-5}$

How do I calculate $\ln (2)$ by Taylor series with error at most $10^{-5}$? I use Maclaurin: $$ \frac{(1+c)^{-(n+1)}\cdot {{(-1)}^n}}{n+1}<\frac{(-1)^{n}}{n+1}<\frac{1}{n+1}<10^{-5} $$ ...
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Proving $\frac{1}{\sqrt{1-x}} \le e^x$ on $[0,1/2]$.

Is there a simple way to prove $$\frac{1}{\sqrt{1-x}} \le e^x$$ on $x \in [0,1/2]$? Some of my observations from plots, etc.: Equality is attained at $x=0$ and near $x=0.8$. The derivative is ...
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1answer
100 views

Finding a Matrix of Rank 10 using Taylor Expansion

For $-1\leq t_i\le 1$ and $1\le i\le n$, we have an $n\times n$ matrix $A$ such that $$A_{ij}=\exp(t_it_j)$$ Now, how can we use the Taylor expansion of $e^x $ to find a $\text{rank }10$ matrix ...
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1answer
25 views

Find the Taylor series and evaluate at $f^{39}(0)$

$$e^{-x^2}$$ I've had a hard time understanding power series since as long as I can remember. To my understanding, the question is asking me to write out the terms in the formula for Taylor series, ...
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1answer
35 views

Cosine of matrix and matrix of cosines

Suppose I have cosine of the matrix $$ \tag 1 \cos\left( \begin{pmatrix} a & b \\ c & d\end{pmatrix}\right) $$ May I write it in a form $$ \tag 2 \begin{pmatrix} \cos(a) & \cos(b) \\ ...
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Let g(x) be the Mclaurin's expansion of sin(2x). If error is atmost $\frac{1250. 10^{-4}}{3} $ for x $\in$ $ [0,\frac{1}{2}]$

Let g(x) be the Mclaurin's expansion of sin(2x). If error is atmost $\frac{1250. 10^{-4}}{3} $ for x $\in$ $ [0,\frac{1}{2}]$ . Then minimum number of non zero terms in g is A.2 B.3 C.4 D.5 I ...
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Over an integral arising from Kepler's problem [also: generally useful integral, NOT DUPLICATE!]

This post might appear as a duplicate of the following: Over an integral arising from Kepler's problem [also: generally useful integral] So recalling quickly: $$\Phi(\epsilon) = ...
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1answer
504 views

Taylor Series Theorem

So I see the argument presented in taylor series, that $$\sum c_n (x-a)^n = \sum \frac{f^{(n)}(a)}{n!} (x-a)^n$$ or $c_n = f^{(n)}(a)/n!$ if $x=a$ the question is, since the above only holds when ...
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1answer
42 views

Calculate $\sin \frac \pi {10}$ with error bound of $10^{-4}$

I know there are similar posts about this question, but I have read them and it's still not clearly for me. I have to calculate $\sin \frac \pi {10}$ with error bound of $10^{-4}$. I know I have to ...
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Taylor series with mixed derivative

let $f: \mathbb{R}^2 \rightarrow \mathbb{R}: (x,y) \rightarrow f(x,y)$ a continuous, derivable function. Let $u := (x,y)$. Now, if you have $f(x+a,y+b) $with $a, b \in \mathbb{R}$ (and close enough to ...
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1answer
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Give the Taylor series for the following $f(z)$; also, find $f^{(100)}(0)$

$$e^{3z}$$ I'm not sure how to approach this complex number problem. I know that $$1+3 x+\frac{9 x^2}{2}+\frac{9 x^3}{2}+\frac{27 x^4}{8}+\cdots$$ is true for $e^{3x}$, but how does this apply to ...
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Convert $f(x)=(\cos(x))^3$ to powers of x and find if converges.

I started out by writing the Taylor series for $x_0=0$ (Maclaurin series) of $f(x)=(\cos(x))^3$. If my calculations are correct $$f(x)=1-\frac{3x^2}{2!}+\frac{21x^4}{4!}-\frac{183x^2}{6!}+...$$ and ...
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Series expansion: $1/(1-x)^n$

What is the expansion for $(1-x)^{-n}$? Could find only the expansion upto the power of $-3$. Is there some general formula?
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1answer
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Comparing a series expansion to polynomial regression

So I don't have a great background in mathematics but I have a quick and hopefully simple question for you guys. I'm a graduate student and I'm doing some polynomial regression on some thermodynamic ...
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1answer
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How to use given permissible error.

If Taylor series expansion of $\cos(x)$ is restricted to only first two terms and the permissible error is$ .54×10^{-2}$, then $x$ can be at most. Should i use the permissible error as this ...
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Sufficient conditions for applying Taylor theorem

Consider a real-valued function $f:\mathbb{R}\rightarrow \mathbb{R}$. Is assuming $f(.)$ twice differentiable at $a \in \mathbb{R}$ enough to apply the Taylor Theorem stating $$ ...
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33 views

Order of $x^{1/x}-1$ as $x \to \infty$

Order of $x^{1/x}-1$ as $x \to \infty$ Answer been given: $x^{1/x}-1=O(\frac{\ln x}{x})$ My attempt is to expand the equation: $x^{1/x}=exp(\frac{\ln x}{x})$ I am unable to proceed further, any ...
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Expand $\sqrt{x(1-x)}$

I am trying to expand $\sqrt{x(1-x)}$ in order to find the order of it as $x\to0$. Using Taylor expansion to find the derivative of the equation as $x\to0$ gives each item $0$. Then how do we expand ...
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Using Taylor's Theorem to show that $\ln(1 + x^2) \leq x^2$

Can we show that if $\operatorname{abs}(x) \lt 1$, then $$\ln(1+x^2) \leq x^2\;,$$ using Taylor's Theorem? I am thinking of expanding it about $x=0$ but I got something like $$f(x) = -x^2 + ...
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Is it possible to very roughly approximate a Taylor Series expansion? [closed]

Say I have the following equation: $$ y=-0.000001x^6+0.00001x^5+0.0001x^4+0.001x^3-0.01x^2-0.1x+1 $$ Is it ever possible to find a Taylor polynomial that is similar enough (but not exactly the ...
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Solving $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $

I would like to work out the result of $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $. Here, $t, i, N_i, m_i$ are positive integers. My effort: $$ \Pi^t_i 2 m_i \left(N_i!\right)^{m_i} \implies (2 m_1 ...
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1answer
39 views

Using Lagrange Remainder to find the approximation of $\sqrt(8)

I'm looking for an approximation of $\sqrt 8$ with an approximation of $10^{-4}$. It was given that $\sqrt 8 =3\sqrt\frac{8}{9}$ so I set up a general series for $ \sqrt{1+x}^\frac{1}{2} $ around zero ...
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1answer
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Expansion of $x^{-1/2}$ at $0$

Consider the function $f(x) = x^{-1/2}$ on the non-negative real line. The point $z=0$ is 'distinguished' because it is the boundary of the domain of $f$, and because $f$ has a pole there. It seems ...
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MacLaurin series for $9\sec(3x)$

A question I've been given asks me to find the first 3 non-zero terms of the MacLaurin series for the function: $y = 9sec(3x)$ Looking at old questions on this forum, I think that this is supposed to ...
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Find Taylor series for function and use it to calculate sum of a series. [duplicate]

So i need in finding the correct Taylor series(expansion) for the following function and use it in calculating the sum of series: $$f(x)=\ln(\frac{1-4x^2}{1+4x^2})$$ ...
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699 views

What's wrong with this use of Taylor's expansions?

I'm trying to find the value of the following limit: $$ \lim_{x \to 0} \frac{x^2\cos x - \sin(x\sin x)}{x^4} $$ Which I know equals to $-\dfrac13$. I tried to do the following: $$ \lim_{x \to 0} ...
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3answers
82 views

What is the Taylor series of $\ln \frac{1-x^2}{1+x^2}$?

What is the Taylor series of $$\ln \frac{1-x^2}{1+x^2}$$ ? I started by evaluating the first derivatives but the more I go, the more complicated they are and I can't identify a pattern.
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1answer
21 views

Why $T_{\infty}\sin(x^2)$ could be written as $x^2+o(x^4)$?

I am confused about the little-oh notation in the Taylor series. As we know, $T_{\infty}\sin(x)=x - \frac{x^3}{3!}+\frac{x^5}{5!}+\cdots+(-1)^n\frac{x^{2n+1}}{(2n+1)!}+\cdots$ By substitution, I got ...
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Taylor series for $f(x)=\cot(x)$

Trying to expand $f(x)=\cot(x)$ to Taylor series (Maclaurin, actually). But I keep "adding up" infinities when using the formula. (Because of $\cot(0)=\infty$) Could you perhaps give me a hint on how ...
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Maclaurin Series of $e^{-x^2}$

The question is: Find the first 3 non-zero terms in the MacLaurin series for the function: $$y =e^{-x^2}$$ I have been told to simply substitute the $-x^2$ into the standard MacLaurin series for ...
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1answer
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Looking to have my worked check on a calculus series question

I am trying to determine if the Taylor series of $f(y) = y^{-\frac{1}{3}}$ about $y=1$ converges absolutely at $y = 2$. I am calculating the Taylor series as $$f(y) = 1 + a_1 (y-1) + a_2 (y-1)^2 + ...
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38 views

Taylor expansion of $\cosh^2(x)$

I have a question which I'm troubling to solve. I've been given the following, a Taylor expansion of $\cosh(x)$ around $x=0$; $$\cosh(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}.$$ Now, using that ...
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1answer
88 views

Conditions of the Taylor Theorem

I'm confused on the assumptions behind the Taylor Theorem because I found different versions of them across several books. Consider the function $f:\mathbb{R}\rightarrow \mathbb{R}$ (1) If and only ...
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LTE for the Cahn Hilliard Equation

I'm trying to Calculate the LTE of a first order finite difference (central difference) scheme for the following 1D equation: $u_t = \frac{\partial^2}{\partial x^2}(u^3 - u - \frac{\partial^2 ...
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Find the value of $n$ such that the Maclaurin polynomial error is within a bound

Let $T_n(x)$ be the $n^{th}$ Maclaurin polynomial for $f(x) = e^x$. Use the error formula to determine a value of $n$ so that $\lvert T_n(2)−e^2\rvert < 10^{−4}.$ I haven't seen a problem like ...
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Definite integral of $e^{x/2}$ using Maclaurin polynomial

My professor asked us to find the 3rd degree Maclaurin polynomial of $e^{x/2}$ which I found to be $$1 + \frac{x}{2} + \frac{x^2}{8} + \frac{x^3}{48}$$ I do know that that the series for ...
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What is the Lagrange remainder in a Taylor series expansion

I know what a Taylor series expansion is and I know how to find the Lagrange remainder but what does it mean intuitively? I need an explanation of what the Lagrange remainder represents in terms of ...
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Calculate $\lim_{x \to 0} \frac{\sin(\sin x)-x(1-x^2)^\frac{1}{3}}{x^5}$ by Taylor's theorem.

I have to calculate : $$\lim_{x \to 0} \frac{\sin(\sin x)-x(1-x^2)^\frac{1}{3}}{x^5}$$ by using Taylor's theorem. I know that :$$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...
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How to find the value of $x$ that $T_{\infty} f(x)$ convert to $f(x)$?

I am not sure whether I am right about the general solution to these questions. I think Taylor series convert to the original function only when $\lim_{n \rightarrow \infty}R_{n}f(x) = 0$ However, ...
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Clarification of notation in multivariate taylor expansion

I'm reading the book "Numerical Optimization" by Nocedal and Wright and on page 14 of the book they present a form of the multivariate Taylor theorem which I find to be a bit peculiar. It is stated ...
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What is a power series representation of $\frac{1-x^2}{1+x^2}$

I got so far as to rewrite to this: $(1-x^2)\cdot \frac{1}{1-(-x^2)}$ so that I can write the power series as follows: $$(1-x^2)\cdot\sum_{n=0}^\infty(-1)^n \cdot x^{2n}$$ But how can I bring the ...
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3answers
58 views

Hundredth Derivative From Taylor's polynomial for $\frac{x^2}{1+x^4}$

I'm trying to solve this problem: Find the hundredth derivative (at $x=0$) from Taylor's polynomial for $\dfrac{x^2}{1+x^4}$. I keep getting the wrong answer; can someone help? I have tried ...
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163 views

How to find all roots of the quintic using the Bring radical

Finding one root $x_1$ of the quintic equation $x^5 + x = -a$ by using the Bring radical is described on Wikipedia. The root is $x_1 = -a +a^5 -5a^9+35a^{13}+ \ldots$ , and it is found by reversion ...
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1answer
482 views

Taylor expansion, integration by parts, and the integration of dt.

So my notes say, for a continuous function we have $$ \int_a^x f'(t)dt = f(x) - f(a) \tag 1 $$ which I understand. So re-arranging gives. $$ f(x) = f(a) + \int_a^x f'(t)dt \tag 2 $$ or $$ f(x) ...
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1answer
37 views

Taylor series for $\frac{1}{z-5}$ at $z=i$

I need to find Taylor series for $\frac{1}{z-5}$ at $z=i$ My attempt: $$f(i)=\frac{1}{i-5}\\ f'(i)=-\frac{1}{(i-5)^2}\\ f''(i)=\frac{2}{(i-5)^3}\\ f'''(i)=-\frac{6}{(i-5)^4}\\ \Longrightarrow ...
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1answer
473 views

Maclaurin series for $\sin x \cos x$

What is the Mauclarin series for $\sin x \cos x$? I would think that you could just multiply out the representation for $\sin x$ with the representation for $\cos x$, but that's apparently wrong. If ...
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1answer
114 views

Radius of convergence for Taylor series?!

Given is: $f(x) = \frac{\sin x}{x} $ I need the Taylor series in $a = 0$, so: $$T(x,0) = \frac{1}{x} \sum_{n=0}^\infty ((-1)^n* \frac{x^{2n+1}}{(2n+1)!} ) = \sum_{n=0}^\infty (-1)^n * ...