# Tagged Questions

Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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### Power Series Solution for Differential Equation of Shifted Exponential Function

I am trying to write a shifted exponential function as a power series. I am aware of the power series definition of the exponential function, i.e. $e^{-x}=\sum_{n=0}^{\infty}\frac{(-x)^{n}}{n!}$. ...
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### Deciphering the theorem of perfect powers

Can someone help me understand what happened to this equation from the paper entitled Perfect Powers With All Equal digits but one...I don't understand the part when it lets a and c not equal to zero....
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This may be a strange question, but I've not found anything about this. Well, anyone can observe that both $$\cos(z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}z^{2n}$$ and $$\sin(z)=\sum_{n=0}^{\... 1answer 48 views ### Power Series in Two Variables and Radius of Convergence Let \alpha > 0, \beta > 0, and assume that the power series with real coefficients $$\sum_{n,m = 0}^{\infty} a_{n,m} x^{n} y^{m}$$ is absolutely convergent for ... 3answers 64 views ### Why does changing the center of a geometric power series change the interval of convergence? I know that the interval of convergence of the geometric power series$$\sum_{n=0}^\infty x^n=\frac{1}{1-x}$$is (-1,1). Why is it that if I do the following manipulation$$\frac{1}{1-x}=\frac{1}{...
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Is there a formula to multiply many series (More than two) using the Cauchy product? If there isn't, please tell me how I can write this formula $\left( \frac{1}{a-e^x} \right) ^{n+1}$as the ...
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### Show that $\sum (-1)^n x^{(2^n)}$ has no limit as $x \uparrow 1$

Show that the following limit does not exist: $$\sum_{0}^{\infty} (-1)^n x^{(2^n)}\text{ with }x \uparrow 1$$ I tried setting $$f(x) = x - x^2 + x^4 - x^8...$$ then $$f(x) = x-f(x^2)$$ then the ...
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### How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
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### Tight bounds on the sum of the $j$th power of the first $n$ natural number?

What are the tight bounds for $S_{n,j}=\sum_{k=1}^n k^j$? Where $O(j)=O(n^3)$.
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### Problem calculating the sine of a matrix

Given the matrix $A=\begin{pmatrix}-\frac{3\pi}{4} & \frac{\pi}{2}\\\frac{\pi}{2}&0\end{pmatrix}$, I want to calculate the sine $\sin(A)$. I do so by diagonalizing A and plugging it in the ...
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### Is it possible to have a power series for arctan(x) centered at 1?

My Calculus 2 professor referenced that such a series is impossible, but why? I understand how to properly find the power series of arctan(x) centered at 0.
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### How to determine sum of an alternating power series and to prove that sum is positive

I am working on a problem involving an alternating power series as follows: $$\sum_{i=0}^{a-2} (-1)^{a+b-i-2}(a+b-i-1)x^{a+b-i-2}$$ $a$ and $b$ is constant with $0<x<1$ I would like to ...
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### Sum of a converging series having Error function with a polynomial

I am struggling to find the sum of the following series: $k\sum\limits_{z=1}^{\infty} \frac{(z+1)^2}{4} . erfc(az)$ where $k$ and $a$ are known parameters and $erfc(x)$ is the complementary error ...
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### What does it mean intuitively for a Taylor Series to be centered at a specific point?

I understand what a Taylor series is and how to find the Taylor series of a function. However I do not understand intuitively what it means to find a Taylor series for a specific function, centered at ...
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### Series $\frac{x^{3n}}{(3n)!}$ find sum using differentiation

Find sum of the series $$\sum_{n=1}^{\infty}\frac{x^{3n}}{\left(3n\right)!}$$ using differentiation. So far I found that $$S(x)+1=S'''(x)$$ but it does not help. Then I tried different interesting ...