# Tagged Questions

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### Taylor series of the function $f(x) = (1+x) ^{\frac{1}{x}}$

Good night!! I got this problem and I'd like to find all the mistakes in my statement. This is my prblem. Find the binomial coefficients of $f(x) = (1+x)^{\alpha}$, with $\alpha \in \mathbb{R}$, and ...
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### Finding Laurent series where given annulus is not in a singularity

I'm given a problem where I need to calculate the Laurent series of $f(z)$ inside the given annulus $$f(z) = {1\over z^3(z-1)}; \quad 1 < |z| < 2$$ From online resources(videos, notes) I ...
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### Taylor polynomial of $\sin(x)$

It is asked to construct the Taylor Polynomial $p_n$ (polynomial of order n) of the function $\sin (x)$, defined in $(-1,1)$ around 0. Also, I need to decide if $p_n$ uniformly converges to sine in ...
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### Estimate Interval of Validity of $1-\frac{x^2}{2}$ for $\cos(x)$

I have been struggling with the following problem and was wondering if anyone could provide some insight or suggestions: Use $1-\frac{x^2}{2}$ as an approximation to $\cos(x)$, with an error not ...
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### Sine taylor series

I'm pretty convinced that the Taylor Series (or better: Maclaurin Series): $$\sin{x} = x -\frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$ Is exactly equal the sine function at $x=0$ I'm also pretty sure ...
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### exp(x) for imaginary numbers

Well, I know how to get the $e^x$ function polynomial expansion, but how do I know that this is also valid for imaginary numbers, like $i\pi$? I know that the ...
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### Evaluate $I=\int_{\gamma (0,1)}\frac{zdz}{(z^2-4z+1)^2}$ using Taylor's Theorem

I've shown that $f(z)=\frac{z}{(z^2-4z+1)^2}$ is holomorphic apart from at points $\alpha=2-\sqrt3$ and $\beta=2+\sqrt3$ and that the talyor coefficient of $g(z)=\frac{z}{(z-\beta)^2}$ centred at ...
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### Help find the MacLaurin series for $\frac{1}{e^x+1}$

What is the MacLaurin series up to $x^4$ for $\frac{1}{e^x+1}$? My Attempt: \begin{align} \frac{1}{e^x+1} &=(1+e^x)^{-1} \\ &\approx 1 -e^x+(e^x)^2-(e^x)^3+(e^x)^4 \\ \end{align} Since ...
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### Taylor series convergence

$$f(z)=\int^z_0 \frac{\zeta-\sin(\zeta)}{\zeta^2+4} \, d\zeta$$ I am supposed to find the convergence radius of its Taylor series at point $a=2$. I can find the radius in simple cases by finding ...
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### Order of accuracy of the following approximation?

This is a computational analysis class, the answer shouldn't be complicated. Given the ODE: $$\begin{cases}y'=f(t,y)\\y(t^0)=y^0\end{cases}$$ What would be the order of accuracy (using Taylor ...
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### What is the difference between Taylor series and Laurent series?

Can someone intuitively describe what is the difference between Taylor series and Laurent series? Also, what is the most general formula for both?
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### Finding Taylor approximation for $x^4e^{-x^3}$

I'm trying to find Taylor approximation for the function: $$x^4e^{-x^3}$$ I started taking the first, second, third, etc. derivatives but the expression for it seems to explode with terms. I was just ...
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### Why is$(1+\frac{1}{n})^n=e$ when n goes to infinity? [duplicate]

Why is $\lim\limits_{n\to\infty}(1+\frac1n)^n=e$? I think it involves $\sum\limits_{n=0}^\infty\frac1{k!}=e$ but not sure how to get from one to the other.
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### Numerical analysis Taylor's method question: Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0.5]$.

Let $f(x)=\tan^{-1}(x)$ Let $P_n(x)$ be the $n$th Taylor polynomial for $f(x)$ about $x_0=0$ Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0.5]$. Is ...
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### Taylor expansion with random variables $\frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}$

In Einsteins theory of relativity the kinetic energy of an object is given by the following formula $$E_k = \frac{mc^2}{\sqrt{1-\frac{v}{c}^2}} - mc^2$$ where m is mass of the object at rest v iss ...
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### Geometric Series into Maclaurin Series

Expand 1/(1+x) into Maclaurin Series I found f(0)=1, f'(0)=1, f''(0)=2!, f'''(0)=3! and so on Therefore f^(k)(0)=k! so would the series centered at 0 be equal to x^k ? Just want to check to see if I ...
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### Short way? Write taylor's formula $f(x,y)=e^{xy}$ $a=(0,0)$ and $p=4$

Write taylor's formula $f(x,y)=e^{xy}$ $a=(0,0)$ and $p=4$ Does there exist any short way? I have to calculate all partial dervatives. Is it?
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### A problem related to mean value theorem and taylor's formula

I guess I need to use Taylor's formula and the mean value theorem. I have no idea except for them. Note: honestly, this is not homework. I am studying by myself. Suppose that ...
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### About Hessian of distance function

I'm studying the comparison Hessian theorem and I not understand the following: Let $(M, \langle\ ,\ \rangle)$ be a complete Riemannian manifold. Given $o\in M$, define $r=dist(o, \cdot)$. Then, for ...
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### Rearranging power series expansion to get parameter on denominator

How can we rearrange $$T=\dfrac{k V+g}{gk}\bigg(kT-\dfrac{1}{2}k^{2}T^{2}+\dfrac{1}{6}k^{3}T^{3}\bigg),$$ to get $$T=\dfrac{2V/g}{1+k V/g}+\dfrac{1}{3}k T^{2}$$ ?
My text, as many others, asserts that the proof of Lagrange's remainder is similar to that of the Mean-Value Theorem. To prove the Mean-Vale Theorem, suppose that f is differentiable over $(a, b)$ ...
### How to evaluate binomial coefficients when $k=0$ and $1\geq|n|\geq0$
So I am doing some taylor series which boil down to the geometric series, for which I then need to evaluate various binomial coefficients. I have always used $n\choose k$ $=\frac{n!}{(n-k)!k!}$ to do ...