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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71 views

Calculating the power series expansion about $\pi/2$ of $g(z)=\tan(z/2)$

Calculate the power series expansion about $\pi/2$ of $g(z)=\tan(z/2)$. Now calculate the expansion about $0$. I'm having trouble doing this. I'm not even sure which is the best way to approach ...
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4answers
70 views

power series for square root matrix

Suppose I have a matrix of the form $$U\ =\ (I+z\thinspace X)^{\frac{1}{2}}$$ where $I$ is the $n\times n$ identity matrix, $z\in\mathbb{C}$ and $X$ is a $n\times n$ arbitrary complex matrix with ...
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2answers
58 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
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31 views

Somehow “mirroring” the Taylor-expansion of some $g(x)$

In proceeding with an older discussion of a summation-procedure for divergent series using the matrix of Eulerian numbers I came to a rather general formulation but cannot/do-not-know-how-to make ...
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1answer
43 views

Moving Center of Power Series

Given a power series: $$\lim_{N\to\infty}\sum_{k=0}^N A_k (z-a)^k$$ I expand the powers: $$\lim_{N\to\infty}\sum_{l=0}^N(\sum_{k=l}^N A_k \binom{k}{l}(-1)^{k-l}a^{k-l})z^l$$ But here I face the ...
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81 views

Is the completion of $(k[x,y]/f)_\mathfrak{m}$ isomorphic to $k[[x]][y]/f$?

Let $k$ be an algebraically closed field and let $f\in k[x,y]$ be an irreducible polynomial with no constant term that is not a polynomial in $x$ alone. Is it the case that the completion of the ...
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1answer
180 views

Introduction to Analysis: Multiplication Theorem for Series

I've been stuck on this problem over the weekend so I decided to ask for some direction. The problem reads: "The multiplication theorem for series requires that the two series be absolutely ...
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62 views

Solving ODE with negative expansion power series [duplicate]

I am solving a series of ODE, such that each DE is equal to some degree of term that I'm expanding to. For instance, one DE is this: $\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$ ...
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1answer
82 views

Modelling ellipsoid-like surface with equations

In my numerical simulation, I would often have to generate the failure surface of some materials. Often, the failure surface will take a ellipsoid-like ball such as this: Of course, the surface ...
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2answers
145 views

Identity with Bernoulli numbers: $\sum\limits_{k=1}^{n}k^p=\frac{1}{p+1}\sum\limits_{j=0}^{p}\binom{p+1}{j}B_j n^{p+1-j}$

How I can prove that $$\sum_{k=1}^{n}k^p=\frac{1}{p+1}\sum_{j=0}^{p}\binom{p+1}{j}B_j n^{p+1-j},$$ where $B_j$ is the $j$th Bernoulli number? I hope to find the answer. Thanks for help.
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62 views

What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$

For a nonnegative integer $b$, and $|x|<1$, what is the function given by the power series $$ \sum_{n=0}^\infty \binom{b+2n}{b+n} x^n. $$ For $b=0$, this post shows $$ \sum_{n=0}^\infty ...
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1answer
225 views

using power series expansion to find a holomorphic function which solves a differential equation

Using power series expansions, find a function $f$ which is holomorphic on the unit disk $D:=$ {$z\in\mathbb C:|z|<1$} and solves the differential equation $(1-z^2)f''(z)-4zf'(z)-2f(z)=0$ for ...
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1answer
1k views

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
111 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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1answer
48 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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1answer
454 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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1answer
64 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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4answers
71 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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197 views

Uniform Convergence: Complex Analysis

To show that $f_k(z) = \frac{z^k}{k}$ converges uniformly for $|z| < 1 $ and that $f'_k(z)$ does not converge uniformly for $|z| < 1$, what must be done? What other things can be said about the ...
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4answers
204 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 = ...
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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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1answer
111 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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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 ...
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1answer
91 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 ...
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92 views

Is there any way to create (a closed form for) this power series/generating function?

There is a fairly simple pattern to it. $$1 y + $$ $$(1 + 1x)y^2+ $$ $$(1+1x+1x^2 + 1x^3)y^3 + $$ $$(1+1x+\dots+1x^7)y^4 + $$ $$(1+1x+\dots+1x^{15})y^5 + $$ $$\dots$$ Does anyone know of a way ...