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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5
votes
3answers
410 views

Compositions of $n$ with largest part at most $m$

This is a problem from Stanley's Enumerative Combinatorics that I'm failing at a bit (lot): Let $\bar{c}(m,n)$ denote the number of compositions of $n$ with largest part at most $m$. Show that ...
0
votes
1answer
112 views

Radius of convergence - ratio test for power series/real numbers

Having trouble with when to apply the ratio test for power series and when to apply the ratio test for real numbers. For example, find radius of convergence of these.... $\sum_{n=0}^{\infty}(-1)^n ...
6
votes
1answer
200 views

Existence of a power series converging non-uniformly to a continuous function

I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that: $f$ converges and is continuous on the closed unit disk $D$ and the series $\sum_n a_n z^n$ does not converge ...
2
votes
1answer
198 views

Hermite's equation of order $\alpha$

Show that the general solution of Hermite's equation of order $\alpha$: $${y}''-2x{y}'+2\alpha y=0$$ $$is$$ $$y(x)=c_{0}y_{1}(x)+c_{1}y_{2}(x)$$ where $y_{1}(x)$ and $y_{2}(x)$ are power series ...
1
vote
2answers
122 views

Finding the $x^n$ coefficient of the power series $\sum\limits_{n=0}^\infty\frac{x^{2n+3}}{n!}$

I have a practice test question that asks: Given the following Maclaurin series representation, $$\sum\limits_{n=0}^\infty\frac{x^{2n+3}}{n!}$$ what is the coefficient of $x^n$? I have the ...
2
votes
1answer
52 views

Residue of a 1-form in a Riemann Surface does not depend of the chart

Let's suppose that $X$ is a Riemann Surface, $\omega$ is a meromorphic 1-form in $X$ and let $p$ be a pole of $\omega$ of order $M$. I want to show that the residue of $\omega$ at $p$, defined by $$ ...
1
vote
1answer
127 views

Determining power series for $\frac{3x^{2}-4x+9}{(x-1)^2(x+3)}$

I'm looking for the power series for $f(x)=\frac{3x^{2}-4x+9}{(x-1)^2(x+3)}$ My approach: the given function is a combination of two problems. first i made some transformations, so the function looks ...
2
votes
1answer
137 views

What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc.

I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful ...
0
votes
1answer
483 views

Expansion of power series for $\frac{\ln(1-x)}{1+x}$

My Problem is to expand $f(x)=\dfrac{\ln(1-x)}{1+x}$ into a power series. My Approach: from looking onto the Graphs of this function, i know, for rising x the y is falling towards Zero, without ...
2
votes
3answers
3k views

Finding the explicit formula for a recursive sequence, using power series

The Task is to find the explicit expression for the given recursive sequence with the help of power series. Given: $a_{0}=0,\ a_{1}=1 \quad$ and $\quad a_{n}=5\cdot a_{n-1} -6\cdot a_{n-2}\quad $ ...
3
votes
1answer
114 views

Is $\pi$ to do with circles or power series?

To get straight to the point: is $\pi$ defined as the ratio of the circumference and diameter of a circle, or as the first non-zero root of the power series of $\sin{x}$? If the former, then $\pi$ ...
2
votes
2answers
105 views

Uniqueness of distribution with moments $M_n$ if $\limsup_{n\to\infty} \frac{1}{n}\sqrt[n]{M_n}$ finite

There's a theorem which states that the moments, i.e. $M_n = \mathbb{E}\left(X^n\right)$, of a distribution uniquely identify the distribution if $$ R := \left(\limsup_{n\to\infty} ...
3
votes
1answer
90 views

Can't solve this series…

I need to solve for the closed form of the following series: $$ S_k(x)=\sum_{n=1}^{\infty} \frac {n} {n^2-k^2}x^n $$ I can't seem to get it in terms of any known series. Differentiating, ...
2
votes
1answer
481 views

Radius of convergence of a power series with Bernoulli numbers

Say, we use the definition: Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$ and then derive power series representations of the ...
8
votes
4answers
269 views

When are we (not) allowed to replace $x$ by $ix$?

It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
4
votes
1answer
566 views

Determining the Radius of Convergence for an alternating power series

I have a given alternating power series: $\sum\limits_{n=2}^{\infty}(-1)^{n}\frac{1}{n2^n}\cdot x^n$ and i need to find the Radius of Convergence. I tried myself with this task and i found an answer, ...
1
vote
1answer
519 views

Solving the second order differential equation: $y' =x^2y$ as a power series

What is the proper way to do this problem as a power series? The way I'm doing it, I end up with a very complicated term. how I'm doing it: take the series for $y$ and assume it's of the form ...
2
votes
2answers
102 views

How does one get the Bernoulli numbers via the generating function?

Here is the definition: Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$ I've tried to naively expand $\frac{x}{e^x-1}$ around ...
1
vote
1answer
58 views

Homework: Maclaurin Power Series Help

I'm trying to find the Maclaurin Power Series for $$f(x)=\frac{3x-8}{3x^2+5x-2}$$ but each degree of differentiation gets more complex with no discernible pattern. Any help is appreciated, thanks.
1
vote
3answers
218 views

Expanding $\frac{1}{1-z-z^2}$ to a power series.

How would you expand the analytic function $$\frac{1}{1-z-z^2}$$ to a series of the form $$\sum_{k=0}^\infty a_k z^k \, \, ?$$
4
votes
1answer
103 views

A numeral system built around Dirichlet series, by analogy of how positional numeral systems are built around power series?

For any natural number and chosen base p, the number admits a unique expression of the form $a_np^n + ... + a_2p^2 + a_1p^1 + a_0$, where $a_k < p$ for all k. This property is effectively what ...
2
votes
2answers
207 views

Sine and Cosine Power Series

I have read that sine and cosine can be represented as power series. Power series, as I understand them, are infinite series that can be represented as: $\sum_{j=0}^{\infty} a_j (x-x_0)^j$ where ...
1
vote
1answer
48 views

Points around which one expands and the radiuses of convergence

I'm trying to make sense of the following passage: Let $f(x)=\frac{1}{x+1}$ and $R_0$ the radius of convergence of the Taylor series of $f$ around $x_0=0$, analogously: $R_1$ — around ...
2
votes
4answers
4k views

Radius of Convergence of power series of Complex Analysis

I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be ...
6
votes
2answers
574 views

Continued fraction expansion related to exponential generating function

A recent SciComp.SE Question motivates us to ask for a nice continued fraction expansion of the following Maclaurin series: $$ f(x) = \sum_{n=0}^\infty \frac{B_n\; x^{n+3}}{n! (n+3)} = \int_0^x ...
1
vote
2answers
170 views

find a solution from power series for multiple variable

$3^x4^y = 4,782,969 $ where $x$ and $y$ are integers. What is the value of $y$? Is there any theory to solve this type problem? i have tried to make $4,782,969$ into power series but couldn't. So a ...
3
votes
1answer
46 views

Estimate of a summation

Show that for each $\alpha \in (0,1)$ there exists a constant $C_\alpha$ such that $$ |F_\alpha(x)| \leq C_\alpha |x|^\alpha $$ for all $x \in \mathbf{R}$ where $F_\alpha$ is given as $$ ...
11
votes
3answers
210 views

Could we show $1-(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dots)^2=(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}- \dots)^2$ if we didn't know about Taylor Expansion?

Suppose that humanity haven't discovered Taylor Series Expansion of trigonometric functions or of any function that would help us on this. Which means we are not allowed to replace the given infinite ...
3
votes
1answer
53 views

Struggling to understand a couple of concepts with series

I have two questions: neither of which are homework problems but certainly pertain to my ability to do the homework. The first regards the harmonic series. The question has been answered often here ...
1
vote
1answer
69 views

(When) Is $\sum\limits_{k=0}^\infty(-1)^k(a+bk)^Nx^k$ positive for all $0 < x < 1$?

I have been stuck with this problem for while. So hope someone will give me a hint how to solve it. For $a$ and $b$ are positive real number, is it true that $$ S = \sum\limits_{k = 0}^\infty ...
2
votes
1answer
125 views

Nontrivial expansion of a multivariate power series in form of a single variable series?

I am trying to interpolate a function defined over a three-dimensional real space: $$f: R^3\rightarrow R\\(x,y,z)\rightarrow f(x,y,z)$$ Let assume I have $N_1 N_2 N_3$ points in the space which form ...
2
votes
3answers
86 views

How to prove this equation?

$\displaystyle\frac1N\sum_{k=0}^{N-1}e^{\frac{i2\pi\mu k}N}=\begin{cases}1,&k\mid\mu\\0,&k\nmid\mu\end{cases}$ where $\mu=0,\pm1,\pm2,\dots$ and $N>0$. I hope for the procedure in detail. ...
0
votes
2answers
186 views

Ratio test and the radius of convergence

Let $$ \sum_{n=0}^\infty c_n (z-a)^n $$ be a power series. If the value $$ r=\underset{n\to\infty}{\lim}\left|\frac{c_n}{c_{n+1}}\right| $$ exists (the limit exists and is a real number), it is the ...
2
votes
1answer
118 views

Radius of convergence of $\sum_{n=1}^{\infty} { (n \sin{\frac{1}{n}})^{n} x^n } $

We need to calculate the radius of convergence $R$ of: $$\sum_{n=1}^{\infty} {\left(n \sin{\frac{1}{n}}\right)^{n} x^n }.$$ Here's what I did: $$ \lim_{n\to\infty} { \left| ...
3
votes
1answer
111 views

Sum of power series $\sum_{n\geq0}\frac{n^2}{7^n}$

I'm having trouble with this power series : $$ \sum_{n\geq0}\frac{n^2}{7^n}$$ I have to solve it using differentiation/integration. I guess I have to approach it as $\sum_{n\geq0}n^2x^n$, where $x = ...
1
vote
1answer
54 views

Functions $f, g$ are given. We know that we can expand them into power series around $x_0=0$, they also satisfy: $f(\frac{1}{k})=g(\frac{1}{k})$ …

Functions $f, g$ are given. We know that we can expand them into power series around $x_0=0$, they also satisfy: $f(\frac{1}{k})=g(\frac{1}{k})$ for sufficiently large $k \in \mathbb{N} $. Prove that ...
2
votes
2answers
999 views

May I use the triangle inequality for infinite series?

I have to prove the following statement: Let $\lim_{n\to \infty}r_n=0$. Show that $\forall\varepsilon>0 \ \ \ \exists \, n_0 \in \mathbb N \ \ \ \forall x \in(-1,1):$ $$\left\lvert ...
0
votes
2answers
118 views

Is there a power series that converges to the function $f(x)= \lvert x\rvert$ for all $x$?

Is there a power series that converges to the function $f(x)= \lvert x\rvert$ for all $x$? I am pretty lost on how to even start this.
3
votes
2answers
3k views

Finding the Laplace Transform of sin(t)/t

I'm in a Differential Equations class, and I'm having trouble solving a Laplace Transformation problem. This is the problem: Consider the function $$f(t) = \{\begin{align}&\frac{\sin(t)}{t} ...
5
votes
2answers
63 views

expand $ \arctan\left(\frac{3x+2}{3x-2}\right)$ into pwer series, find radius of convergence (check solution)

I would be grateful if someone could check what I've worked out: $$ f(x)=\arctan\left(\frac{3x+2}{3x-2}\right)\implies f'(x)=\frac{1}{1+(\frac{3x+2}{3x-2})^2}\cdot \frac{3(3x-2)-3(3x+2)}{(3x-2)^2}$$ ...
1
vote
3answers
166 views

Expand into power series $f(x)=\log(x+\sqrt{1+x^2})$

As in the topic, I am also supposed to find the radius of convergence. My solution: $$\log(x+\sqrt{1+x^2})=\log \left ( x(1+\sqrt{\frac{1}{x^2}+1})\right )=\log(x)+\log(1+\sqrt{\frac{1}{x^2}+1})$$Now ...
1
vote
3answers
122 views

power series of a matrix well-defined

I am working on a seminar lecture and have found the following lemma without a proof: Given a convergent power series $f(z)=\sum_{n=0}^\infty a_nz^n$ and a diagonalizable matrix $M$ with diagonal ...
2
votes
4answers
54 views

Find expansion around $x_0=0$ into power series and find a radius of convergence

My task is as in the topic, I've given function $$f(x)=\frac{1}{1+x+x^2+x^3}$$ My solution is following (when ...
4
votes
2answers
419 views

Analytic functions of a real variable which do not extend to an open complex neighborhood

Do such functions exist? If not, is it appropriate to think of real analytic functions as "slices" of holomorphic functions?
2
votes
1answer
340 views

Show that some $C^\infty$ real function is analytic

Consider the function $f:(a,b) \rightarrow \Bbb R $, which is $C^\infty$ class. It is required to show that if the function $f$ satisfies: $\forall_{n\in\Bbb N_+} \forall_{x\in(a,b)} f^{(n)}(x) \ge 0 ...
3
votes
2answers
358 views

$\sum (-1)^n/n$ fails the p-series test, but passes the alternating series test?

P-Series Reference Alternating Series Test Reference $$ \sum_{i=0}^\infty \frac{(-1)^n}{n} $$ This alternating series fails the p-series test because the exponent of n = 1. Yet it seems to pass ...
2
votes
3answers
850 views

Difficulties performing Laurent Series expansions to determine Residues

The following problems are from Brown and Churchill's Complex Variables, 8ed. From §71 concerning Residues and Poles, problem #1d: Determine the residue at $z = 0$ of the function ...
0
votes
2answers
62 views

Show that cosh(2) is between two values.

I'm reviewing for exams and this question has got me stumped: Show that: $3\dfrac{2}{3} \leq \cosh(2) \leq 3\dfrac{2}{3} + 0.1$ I've determined the series form of cosh(x) to be: ...
2
votes
1answer
134 views

Why do power series converge to a function symmetrically?

Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$? The selected answer to the above question says that for a a power series, the interval of convergence for the ...
0
votes
4answers
71 views

Power Series Proof w/ Binomial Coef.

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