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).

learn more… | top users | synonyms

0
votes
1answer
52 views

Prove that $ \frac12 [log(1-x)]^2 = \frac12x^2 + (1+ \frac12) \frac13 x^3+\cdots$ for $-1<x<1$

$\frac12 [log(1-x)]^2 = \frac12 x^2 + (1+ \frac12) \frac13 x^3+ (1+ \frac12 + \frac13) \frac14 x^4+ \cdots$ My attempt: I'm thinking of finding a series which is convergent in $-1<x<1$ and ...
0
votes
1answer
83 views

Derive an explicit formula for a power series

Could anyone help me find an explicit formula for: $$ \sum_{n=1}^\infty n^2x^n $$ We're supposed to use: $$\sum_{n=1}^\infty nx^n = \frac{x}{(1-x)^2} \qquad |x| <1 $$
0
votes
1answer
92 views

Multiplicative Inverse of a Power Series

For a formal power series $$F(x) = \sum p_i x^i$$ a multiplicative inverse of $F$ exists iff $p_0 \neq 0$. The inverse $\sum q_i x^i$ satisfies the recursion $$q_0 =\frac{1}{p_0}\\ q_{n} = ...
0
votes
1answer
521 views

Method for solving ODE with power series

when trying to solve second order linear homogeneous variable coefficient ODEs using a power series method, there seem to be two different general forms cropping up in my notes. The first uses an ...
0
votes
1answer
69 views

Calculate the Radius of convergence of $\sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$

I need your help: Calculate the Radius of convergence of the following: $$ \sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$$ Im new to this subject, so I'd appreciate it if you can add explanations to ...
0
votes
1answer
301 views

Find the first 5 terms of the expansion in a power series

Find the first 5 terms of the expansion in a power series $$y′=xe^{x}+2y^{2}$$ I've got a riccati equation $$ x e^{x}+2y^{2}, y(0)=0$$ After solving: $$y=e^{x}(x-1)+\frac{2}{3}y^{3} - 1$$ And I ...
0
votes
4answers
75 views

Power Series Proof w/ Binomial Coef.

Prove that, for any positive integer k, $$\sum_{n=0}^\infty {{n+k \choose k}z^n}=\frac{1}{(1-z)^{k+1}}, |z| < 1$$
0
votes
4answers
209 views

If $\sum_{n=0}^\infty c_n4^n$ is convergent, is $\sum_{n=0}^\infty c_n(-4)^n$ convergent as well?

Please identify the flaw in my reasoning: $\displaystyle \sum_{n=0}^\infty c_n4^n$ is convergent, so by the ratio test: $\displaystyle \lim_{n \to \infty}\left\vert\frac{a_{n+1}}{a_n}\right\vert = ...
0
votes
3answers
1k views

Approximating cube root function for small values of $x$

How can one show that for small values of $x$, $\sqrt[3]{x+1}\approx1+\frac{x}{3}$?
0
votes
1answer
115 views

Binary relationship between powers and sum of powers.

I want to optimize a function that determines whether a given number $n$ is EITHER (a power of 2) OR (the sum of powers of 2). Using, this answer, it appears that a sum of power of 2s contain at most ...
0
votes
1answer
1k views

Power series expansion

I recently had a problem. I know how to evaluate power series but I cannot seem to find an expansion for $\sqrt{x+1}$. I've tried differentiating it, in order to bring it in reciprocal form but that ...
0
votes
1answer
95 views

How can we take a power series and multiply each term, i.e. $c_n x^n$ by $y^n$?

In other words, given a power series $f(x)$, is there an alternative to taking $\lim_{x\to{x y}}f(x)$? I ask this because I thought that there may be a way to replace the limit by integration, or ...