# Tagged Questions

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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### $f(x)-T_{f,x_0,n}=o(x-x_0)^{n} \implies f(x) \sim T_{f,x_0,n}$?

I'm not sure about this. Consider a function $f$ and it's $n$ degree Taylor Polynomial in $x_0$ $T_{f,x_0,n}$. Considered the remainder function $R(x)=f(x)-T_{f,x_0,n}=o(x-x_0)^{n}$ Can I ...
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### Series around $s=1$ for $F(s)=\int_{1}^{\infty}\text{Li}(x)\,x^{-s-1}\,dx$

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...
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### Inequality for a multivarialbe function?

For fixed $y\in \mathbb{C}^m$ and let $f$ be a fuction defined on $\mathbb{C}^m\times \mathbb{C}^m$ such taht $f(0,y)=1$ and $$\frac{\partial^n} {\partial x^n}f(x,y)=(i)^my^n{}f(x,y)$$ which means ...
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### Connection between properties of taylor series and the function

Assuming I have a function $f(x)$ which at least for some $-R<x<R$ can be expanded in taylor series $$f(x) = \sum_{n=0}^{\infty}c_n \frac{x^n}{n!}$$ are there any known connections ...
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### Properties of the remainder function for Taylor polynomials

Considered $f$ differentiable at least $n$ times in $x_0$ and $P_{n,x_0}(x)$ the $n$ degree Taylor polynomial in $x_0$. Defined the Remainder function $R(x)= f(x)-P_{n,x_0}(x)$ I can't understand ...
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### Edgeworth expansion of the sum of inid random variables?

This question relates to the asymptotic expansion for the distribution of sum of random variables using moments. Edgeworth expansion can be applied when the variables are independent and identically ...
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### Prove that $\cosh^{-1}(1+x)=\sqrt{2x}(1-\frac{1}{12}x+\frac{3}{160}x^2-\frac{5}{896}x^3+…)$

How can we prove the series expansion of $$\cosh^{-1}(1+x)=\sqrt{2x}\left(1-\frac{1}{12}x+\frac{3}{160}x^2-\frac{5}{896}x^3+...\right)$$ I know the formula for $\cosh^{-1}(x)=\ln(x+\sqrt{x^2-1})$...
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### How to show that $\frac{159999}{80000} +\frac{1}{100e^2} <\ln(e^2+\frac{1}{100} ) < 2+ \frac{1}{100e^2}$

I'm trying to show that $\frac{159999}{80000} +\frac{1}{100e^2} < \ln(e^2+\frac{1}{100} ) < 2+ \frac{1}{100e^2}$. I know I should do something with the first order taylor polynomial of $\ln(x)$ ...
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### Finite expansion of this function

I had this result as finite expansion of this function $$\frac{1}{(1-x)^2}$$ to order n in neighborhood of 0: $$1+\sum_{i=1}^{n}{(x^i.(i+1)) }+0(x^n)$$ (where x tends to 0)is it true? And if yes ...
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### External and internal multipole expansion for axisymmetric potential - the region of convergence.

Say, we have a system of electrodes exhibiting symmetry around a certain axis. We know the explicit expression for the potential on the axis $\phi (z)$. We want to find the potential for any point in ...
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### Prove (or disprove) that $\sum_{n=1}^\infty \frac{4(-1)^n}{1-4n^2} x^n = \frac{2(x+1) \tan^{-1}(\sqrt x)}{\sqrt x} - 2$ for $0<x\leq1$

Just like title said, for $0 <x\leq1$, prove/disprove: $$\displaystyle \sum_{n=1}^\infty \dfrac{4(-1)^n}{1-4n^2} \cdot x^n \stackrel{?}{=} \dfrac{2(x+1) \tan^{-1}(\sqrt x)}{\sqrt x} - 2$$ I ...
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### Why does the taylor expansion of a nonlinear system of differential equations exist if it has continuous second order partial derivatives?

My textbook states that for a nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y)$$ The system is locally linear in the neighborhood of a critical point $(x_0,y_0)$ whenever ...
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### Taylor Expansion of Power of Cumulative Log Normal Distribution Function - Show Lagrange Remainder tends to Zero

QUESTION I am looking to find a simplification of the expression below. I have attempted this using the Taylor series. The question then remains if we can show the Lagrange remainder goes to zero. I ...
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### Why is this proof for Taylor's Remainder theorem not correct?

I am not exactly sure on how to post math equations in the question box so I have all my following information on a google document: https://docs.google.com/document/d/1vf20ZyLGQL-...
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### Prove Taylor expansion with mean value theorem

On http://vapour-trail.blogspot.com/2006/03/brief-explanation-of-taylor-series-via.html one can find an hint at how to derive Taylor expansions from the mean value theorem. The process goes as follow....
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### Taylor Series Polynomial Proof using Induction

If $f : \mathbb R \to\mathbb R$ is a polynomial function of degree $n$ with $a \in\mathbb R$. Show that the $n$-th Taylor polynomial $P_{f,a,n}$ of $f$ at $a$ is equal to $f$. I know that I need to ...
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### Finding $w_1,w_2,b$ such that $w_2 \sigma(w_1 x + b) \approx x$

Let $\sigma(z) = 1/(1+e^{-z})$. How can I find $w_1, w_2, b$ such that $w_2 \sigma(w_1 x + b) \approx x$ for $x \in [0,1]$? The hint provided was to rewrite $x = \frac{1}{2}+\Delta$, assume $w_1$ is ...
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### Find $\lim\limits_{x\to 0}(\cos(xe^x)-\ln(1-x)-x)^{1/x^3}$

Find $$\lim\limits_{x\to 0}(\cos(xe^x)-\ln(1-x)-x)^{1/x^3}$$ $$\lim\limits_{x\to 0}(\cos(xe^x)-\ln(1-x)-x)^{1/x^3}=e^{\lim\limits_{x\to 0}\frac{\ln(\cos(xe^x)-\ln(1-x)-x)}{x^3}}$$ Using Taylor ...
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### Different forms of remainder in Taylor series

In the literature, it is common to find different forms of the remainder function in a truncated Taylor series. To name a few: Integral form Lagrange form Cauchy form First, can you tell me any ...
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### What is the 90th derivative of $\cos(x^5)$ where x = 0?

Trying to figure out how to calculate the 90th derivative of $\cos(x^5)$ evaluated at 0. This is what I tried, but I guess I must have done something wrong or am not understanding something ...
I want to find $$\sum\limits_{n = 1}^\infty {{{{{({1 \over 2})}^n}} \over {n(n + 1)}}}$$ The book that has this problem in says to use the Maclaurin series for $(1 - x)\log (1 - x)$. I don't ...