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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Radius of convergence for the exponential function

I'm studying physics and am currently following a course on complex analysis and in the section on analytic functions, the radius of convergence $R$ for power series was introduced. The Taylor ...
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75 views

series look up site

Is there a site for looking up a series to see some of the associated functions. (In the spirit of Encyclopedia of Integer Sequences OEIS.) In particular I am looking for functions related to $ \sum ...
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3answers
112 views

Formula to $\ln$ that holds on interval $x \geq 1$

In the Wikipedia we can find two formulas using power series to $\ln(x)$, but I would like a formula that holds on the interval $x \geq 1$ (or is possible to calculate $\ln(x)$ to $x \geq 1$ with the ...
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32 views

Question about the coefficient of operator

Note that the "coefficient of" operator is an operator that takes the coefficient of the power series. We start with the following: $$ \frac{1}{f(x)+z} - \frac{1}{f(x)} = \sum_{k=0}^\infty ...
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57 views

Use series methods to find solution corresponding to..

Use series methods to find solution corresponding to $a_0 = 1$ for the equation $(x+1)y' - y = 0$ Here is my work. Can someone verify that I have the correct solution: So for my final solution I ...
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30 views

Question about multiplying summations with another summation inside

I have the following: $$ y = \sum_{n=0}^\infty [x^n \sum_{k=0}^\infty (k+1)a_{k+1} P_{n-k}] \sum_{n=0}^\infty x^n[s_n - \sum_{k=0}^n a_{k+1}(k+1)R_{n-k}] $$ I can easily multiply $$ ...
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51 views

Truncation of partitions generating function question

$A (x)$ is the generating function for partitions. $B(x)=\sum_{n=0}^{\infty}b_nx^n $ $$b_n =\binom{\text{number of partitions of }n}{\text{into an even number of parts}}-\binom{\text{number of ...
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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 ...
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98 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: $$\sum_{n=1}^\infty nx^n = \frac{x}{(1-x)^2} \qquad |x| <1 $$
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114 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} = ...
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540 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 ...
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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 ...
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1answer
313 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 ...
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4answers
76 views

Power Series Proof w/ Binomial Coef.

Prove that, for any positive integer k, $$\sum_{n=0}^\infty {{n+k \choose k}z^n}=\frac{1}{(1-z)^{k+1}}, |z| < 1$$
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4answers
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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 = ...
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3answers
1k views

Approximating cube root function for small values of $x$

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