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Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n (x-c)^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex.
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Series solution for $x^2y''-x(x+6)y'+10y=0$
I have to solve this differential equation: $$x^2y''-x(x+6)y'+10y=0$$ by using this method and I am stuck at this step. Please help me to solve it. Here is my attempt: ordinary-differential-equations power-series differential-forms formal-power-series- 11
Finding Laurent series for $f(z)=\frac{4z^2+2z-4}{z^3-4z}$ around $z=2$
Having $$f(z)=\frac{4z^2+2z-4}{z^3-4z}$$ find the Laurent series in $z=2$ the scope of $z$ is $0<|z-2|<2$ here is my approach: $f(z)=\frac{4z^2+2z-4}{z^3-4z}=\frac{1}{z}+\frac{2}{z-2}+\frac{1}{... power-series laurent-series- 3
Solve for the power series of $(1+x)^n$
I'm having a difficulty on solving this equation about power series. I am asked to solve: $(1+x)^n$ and I need to use this equation: $\sum_{n=0}^{\infty} ar^n= \frac{ar^2}{1-r}$ Lastly, I need to find ... calculus sequences-and-series power-series- 1
Power series of $(1+x)^n$ [closed]
Solve for the $n$th term of the power series of $(1+x)^n$. where n=2 (include the notation) calculus sequences-and-series power-series- 1
A power series that converges conditionally for all points on the radius of convergence?
For $a_n\in\Bbb C$ let $$f(z) = \sum_{k=0}^\infty a_n z^n \tag 1$$ be a power series with radius of convergence of 1, and $a_n$ such that the series converges for all $z\in\Bbb C$ with $|z|=1$. What's ... complex-analysis power-series absolute-convergence conditional-convergence- 5,144
Linear Elliptic PDE Variable Coefficients Non-Separating Variables
I am trying to obtain Analytical solutions for the following Linear Elliptic PDE in the dependent variable U(x,y) having variable coefficients. 'x' is a (pseudo) radial coordinate, and 'y' is an ... partial-differential-equations power-series maple elliptic-equations legendre-functions- 11
Test for Convergence $\sum_{n=1}^\infty\frac{7^{3n}}{n!}$ and $\sum_{n=1}^\infty\sqrt{\ln\frac{n+5}{n+2}}$
Test the following series for convergence $$\sum_{n=1}^\infty\frac{7^{3n}}{n!}$$ and $$\sum_{n=1}^\infty\sqrt{\ln\frac{n+5}{n+2}}$$ I need this for studying purposes, I have an exam next week and I am ... sequences-and-series convergence-divergence power-series- 1
Compute $f^{(2020)}(0)$
Problem : Let $$f(x)=\frac{x}{(x+1)(1-x^2)}$$ Then find $f^{(2020)}(0)$. My Attempt : From partial fraction decomposition, $$f(x)=\frac{1}{4(1-x)}+\frac{1}{4(x+1)}-\frac{1}{2(x+1)^2}$$ and, $$\frac{1}... calculus sequences-and-series power-series- 1,873
Conditions and correct interpretation of Borel summation
Hello to the community. In my line of research (theoretical particle physics) it is customary to apply the strategy of Borel summation to infinite power series in order to find closed forms and/or ... real-analysis sequences-and-series complex-analysis asymptotics power-series- 1
Prove there is a single function which is able to be developed to Power series around $x_0 = 0$
I have given: $$f''(x) - 2f'(x) + f(x) = 0$$ $$f(0) = 0$$ $$f'(0) = 1$$ $$find::---- f(x)=?$$. I wanted to try it by assuming that the series $$f\left(x\right)\:=\:\sum _{n=0}^{\infty }\:a_n\cdot x^n$$... power-series- 23
Can I directly substitute a specific complex number into a formal power series equation?
Suppose there is a formal power series equation, such as $$\sum_{i=0}^\infty a_i x^i=\sum_{i=0}^\infty b_i\left(\sum_{j=1}^\infty c_j x^j\right)^i.$$ If there is a complex number $r$ which makes $\sum ... power-series formal-power-series- 323
$\sum _{n=1}^{\infty }\:\frac{1}{n}\cdot x^{n^2}$ - Power series - converge and Radius convergence range.
I know how to play with power series and find Radius and convergence range. But first time I see $\sum _{n=1}^{\infty }\:\frac{1}{n}\cdot x^{n^2}$ with $x^{n^2}$ How do I start? no need for any answer,... power-series- 87
Radius of convergence for $\sum_{n=0}^{\infty}\frac{c_{n}}{1+\left|c_{n}\right|}z^{n} $
Given $$ ROC\left(\sum_{n=0}^{\infty}c_{n}z^{n}\right)=R $$ I used the root test and figured that:$$ \lim_{n\to\infty}\sqrt[n]{\left|c_{n}\left|z\right|^{n}\right|}<1\Rightarrow\left|z\right|\sqrt[... complex-analysis power-series- 590
Exponential Function Expansion but with Double Factorials
Considering: $\sum_n \frac{z^n}{n!}=e^{z}$ I was wondering if there is anything similar for: $\sum_{n:\text{ odd or even} }\frac{z^n}{n!!}$ That is, when you replace $n!$ with $n!!$ and add for all ... combinatorics taylor-expansion power-series exponential-function- 137
$\sum _{n=1}^{\infty }\:\frac{(\frac{1}{12})^n}{n^x+|x^n|}$ - Find convergence ranges.
Given $\sum _{n=1}^{\infty }\:\frac{\left(\frac{1}{12}\right)^n}{n^x+\left|x\right|^n}$, I need to find the range of convergence. Now, I actually found it, I did: $$\sum _{n=1}^{\infty }\:\left(\frac{... convergence-divergence power-series- 87
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