# Tagged Questions

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### Taylor Series expansion and first four terms of $7x^2 e^{-4x}$

As the series I got $$\sum_{n=0}^\infty (-1)^n(4x)^n/n!$$ which I think is right. However, I am not sure how to get the first four non zero terms.
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### Radius and interval of convergence of the power series $\sum 2^{n^2}x^{n!}$?

How to calculate the radius and interval of convergence of the following series: $$\sum 2^{n^2}x^{n!}$$ The formula for the radius is: $$R = \frac{1}{\limsup_{n\to\infty} \sqrt[n]{|a_n|}}$$ or ...
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### Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)}$

Evaluate: $$\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)}$$ I rewrote the sum as $$\sum_{n=0}^{\infty} \frac {1}{4^{8n-7}(8n-7)} - \sum_{n=0}^{\infty} \frac {1}{4^{8n-3}(8n-3)}$$ Now, I ...
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### Find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$

Let $\alpha$ and $\beta$ be two given constants, how to find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$, where $c_0 = {\alpha}^{\beta}$.
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### Prove that $\exp(\log(\frac{1}{1-x})) = \frac{1}{1-x}$

I am trying to prove this directly by comparing the coefficients in the two series rather than using formal calculus. Here is what I have so far, but I think I made a mistake: \begin{align*} ...
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### Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$

This question involves the following homework problem: PROBLEM Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form ...
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### Question about infinitely many times differentiable function.

Could you please give me some hint how to solve this problem: Suppose $f(x)$ is infinitely many times differentiable function on R, $f(0)=f'(0)=f''(0)=0$. Prove : for all $A>0$ exists some ...
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### Given a power series

Let c be a fixed number and consider the power series $\displaystyle\sum_{n=1}^ \infty \frac{c^{n-1}}{n} x^{n}$. a) Determine the convergence radius r for every value of $c \in \mathbb{C}$. In this ...
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### Determine the maximal compact interval such that $\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1} = \arctan(x)$ holds true

The Assignment: Determine the maximal compact interval, such that the following identity holds true:$$\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1} = \arctan(x)$$ Explain your answer and show ...
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### Is there a power series which converges to $f(x) =| x|$ for all $x$?

I'm confused how to solve the following problem: "Is there a power series which converges to $f(x)$ = $\left| x\right|$ for all $x$?" Your help is greatly appreciated. Thanks a lot!
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Use the identity $\cos((k-\frac{1}{2})x) - \cos((k+\frac{1}{2})x) = 2\sin kx \sin \frac{x}{2}$ to show that $S_n:=\sum_{k=1}^{n}\sin kx=\frac{1}{2\sin ... 2answers 62 views ### What are the values ​​for which the series converges? Determining the values ​​of$a$and$b$so that the series$\sum a_n$converges, where $$a_n=\ln n-a\ln(n+1)+b\ln(n+2)$$ 1answer 53 views ### Simple differentiation question that I am unsure about I am in the process of re-learning differentiation and am stuck on this as part of a larger problem. Can you explain to me why when differentiated 4 times this: $$y = \sum_{n=0}^{+\infty} ... 1answer 121 views ### Finding coefficients of a differential equation represented by power series I am studying for a discrete mathematics exam and have gotten stuck on this question: Any function y of a real variable x that solves the diff erential equation:$$\frac{d^4y}{dx^4} -16y =0$$may ... 2answers 42 views ### General case of radius of convergence of a power series Show that if the series \sum_{n=1}^{\infty} a_nx^n has a radius of convergence L = R so the series \sum_{n=1}^{\infty} a_nx^{kn} has radius of convergence L = R^{\frac{1}{k}}. Anyone could ... 2answers 122 views ### Show that \sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2 [duplicate] Show that \sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2. I have no idea to solve this problem. Anyone could help me? 1answer 80 views ### Interval of convergence \sum_1^\infty \frac{2^n}{3n}(x+3)^n$$\sum_1^\infty \frac{2^n}{3n}(x+3)^n$$I do the ratio test to find the radius.$$\frac{2^{n+1}}{3(n+1)}(x+3)^{n+1} *\frac{3n}{2^n (x+3)^n}$$This should reduce down to 2|x+6|< 1 This is ... 1answer 64 views ### Interval of convergance \sum_0^\infty \frac{(2n)!}{(n!)^3}*x^n$$\sum_0^\infty \frac{(2n)!}{(n!)^3}*x^n$$I have no idea how to do this. I tried to write it all out with the ratio test and I get some weird expression that doesn't make sense like$$x * ... 2answers 74 views ### Let$f(x) = \int \frac{x}{1-x^{8}}dx\,$Let$f(x) = \int \frac{x}{1-x^{8}}dx\,$. Represent$I(x)$by a power series$\sum^{\infty}a_{n}x^{n}$.(Find$a_{n}$) What is the radius of convergence of$I(x)$? Two curves are generated by polar ... 1answer 43 views ### 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 $$T=\dfrac{2V/g}{1+k V/g}+\dfrac{1}{3}k T^{2}$$ ? 1answer 230 views ### Radius of convergence of power series (complex) I don't know if my reasoning is right on this exercise: If the power series$\sum a_n z^n$has radius of convergence$R$, which is the radius of convergence of the series$\sum a_n^2 z^n$and$\sum ...
A real power series $\sum_{n=0}^\infty a_n z^n$ has radius of convergence $R$. I am able to prove that for any real number $r>R$, the sequence $|a_n|r^n$ must be unbounded. Must it also tend to ...