# Questions tagged [taylor-expansion]

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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### How to calculate $\sin(37°)$ with a Taylor approximation?

How to calculate $$\sin(37°)$$ with a Taylor approximation accurate to 3 decimal digits? I know it is not a difficult question, but I have no answers of my book and so far I have only determined the ...
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### The process of using taylor series to evaluate limits.

For instance we want evaluate this simple limit using taylor series : $$L=\lim_{x\to 0}\frac{\sin x}{x^5}=\lim_{x\to 0}\frac{x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}+\cdots}{x^5}$$...
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### Proof using taylor series

I am currently trying to solve a quantum mechanics problem in which i need to prove that $e^A e^{-A} = 1$ where $A$ is an operator and the exponent function is defined by a taylor series. However, I ...
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### Integral $x^{-x}\,dx$ using Taylor's?

So I'm good w my comparison tests but is this an elliptical function? $$\int_0^int_12 x^{-x} \, dx$$ Does one use Taylor's integration in this?
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### Verifying equalities of complex functions and series

I have an exercise that I don't know how to solve, I don't even know how to start to solve it: Let ${z \over exp(z)-1} = 1+B_1z+{B_2 \over 2!}z^2+···+{B_n \over n!}z^n+···$ a) Prove that the complex ...
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### Power series for $\sqrt{1 + x}$ [closed]

How can I decompose a function $\sqrt{1 + x}$ into a power series for any x value?
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### Imaginary asymptotics for the digamma function

I often see asymptotics and precise expansion for the gamma $\Gamma$ or the digamma $\psi$ function $\psi$ when the argument goes to $+\infty$, in particular when it stays real (or in a given angle ...
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### Series Expansion of $f(z) =$ $\frac{e^z}{z^3}$

I've come across this problem on my lecture book of Complex Variable Function, Expand $f(z) =$ $\frac{e^z}{z^3}$ into a power series on the $0 < |z| < \infty$ domain so far I've been learning ...
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### Determining partial derivatives via Taylor polynomials

Typical Taylor series problem which has totally perplexed me. The function is $$e^{x_1x_2}\sin(x_1+x_2)$$ and the task is to solve $\partial_1^2\partial_2f(0,0)$ and $\partial_1\partial_2^3f(0,0).$ ...
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### Help with Taylor Polynomial Estimation Solution.

Estimate $e^{0.1}$ to 6 decimal places using a Taylor polynomial about 0. Use error bounding to prove that your estimate is accurate to at least 6 decimal places. Is my solution correct? We know that ...
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### Taylor expansion of exponential around 0 for $x = O(1)$

Let $x$ be some real value that is $O(1)$, find the taylor expansion of $e^tx$ at $0$. The text I am following writes: $$e^tx = 1 + tx + O(t^2x^2e^{O(t)})$$ I'm confused why there is an additional ...
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### MacLaurin Series of $\tan(x)$

I am trying to compute the MacLaurin series of $\tan(x)$. I know this one is $$\tan x=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}$$ And I know how to derive this formula. ...
Theoretically, how should I compare the precisions of the following two approximations? $f(x_2) = f(x_1) + (x_2-x_1)\times df(x_1)$ $f(x_2) = f(x_1) + (x_2-x_1)\times\frac{[df(x_1)+df(x_2)]}{2}$ where ...
Suppose $f: \mathbb{R}^n \to \mathbb{R}$ is smooth at its isolated local minimum $0$ and $f(0)=0$. Further suppose that there is some $N$ such that in any direction $u$, the $k$-th order directional ...