# 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).

3answers
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### Working out $\tan x$ using sin and cos expansion

Using only the series expansions $\sin x = x- \dfrac{x^3} {3!} + \dfrac{x^5} {5!} + ...$ and $\cos x = 1 - \dfrac{x^2} {2!} + \dfrac{x^4}{4!} + ...$ Find the series expansions of the $\tan x$ ...
1answer
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3answers
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### Power series expansion using Taylors Theorem.

So the function $f(x)=3x^2-6x+5$ needs to be written as a power series expansion around $x=a$ and the goal is to show $x=a$ is $f(x)$ for every $a$. So I started off by finding up to the third ...
1answer
22 views

### Help with general power series concept

If f(x) is some general polynomial, what will the power series expansion of f(x) be. Is there a set rule for finding the power series of polynomials.
2answers
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### How can I solve the following differential equation without use power series [closed]

Let we have the following differential equation $$y''-xy'=e^{-x}$$ how can I solve this differential equation without use power series
0answers
70 views

### How to solve $1 = \sum_{p \text{ prime}} x^{-p-1}$?

As the title says, I am trying to solve the equation $$1 = \sum_{p \text{ prime}} x^{-p-1}$$ and I'm not really sure where to begin. I got this from an exercise in a book and apparently there is a ...
2answers
47 views

### How can I solve the following differential equation [closed]

How can I solve the following differential equation by power series near the point $z=1$ $$(z^2-2z+2)w''+2(z-1)w'=0$$ Then I have to find the radius of convergence of the solution
3answers
53 views

### How to solve power series expansions.

The function is $f(x)=1/(1-x)$ and it asks to find a power series expansion expanded around $x=a$, which would be the general expansion as well as around $x=0$ and $x=2$.
4answers
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### Power Series representation of $\frac{x^3}{(3x+4)^2}$

How do you do this? I have an exam in 2 hours and I know this type will be on it and I have no clue. We were taught to base it off the power series of $x^n$
2answers
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### Estimate on rate of growth of a power series

Given two sequences $(a_k),(b_k)$ with $a_k\geq0,b_k>0$ such that the power series $\sum_{k=0}^\infty a_k b_kr^{k}$ and $\sum_{k=0}^\infty a_kr^k$ converge for each $r>0$. My question now is: ...
1answer
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### An entire function is a polynomial iff the Taylor expansion around $0$ converges uniformly

Let $g:\mathbb{C} \to \mathbb{C}$ an entire function. Prove that the Taylor expansion around $0$ converges uniformly in all $\mathbb{C}$ if and only if $g$ is a polynomial. 1/2 PROOF I think I have ...
1answer
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### Multiplying and factoring in Formal Power Series

I'm working with some formal power series in my homework. Somewhere in the middle of my hw problem I reach a point where I would really like to factor, but I'm not sure if I can. Suppose $F_k$ ...
2answers
79 views

1answer
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### General solution of $(1-x^2)y''-2y=0$ about $x_0=0$?

I've expanded this differential equation as a series to obtain the recurrence relation $$a_{n+2}=\frac{a_n(n^2-n+2)}{n^2+3n+2}.$$ I don't know how to find $a_n$ in terms of $a_0$ and $a_1$ so that I ...