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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Prove Alternating Series Approximation

Prove if $S=\sum_{n=1}^{\infty}a_{n}$ is an alternating series with $\left | a_{n+1}\right | < \left | a_{n} \right |$, and $\lim_{n\to\infty}a_{n}=0$, then $\left |S-(a_{1}+a_{2}+\cdots+a_{n}) ...
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2answers
47 views

Is there a power series expansion of the Signum function?

I would like to define a linear operator as the sign of a another operator. But to use it I would need to expand it out as a power series. Roughly, I'm wondering if something like this ...
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1answer
15 views

Functional equation, inverse w.r.t composition, which function gives identity?

Given a function $$f:\mathbb{C}\to\mathbb{C},\qquad z\mapsto \frac{1-2z-\sqrt{1-4z}}{2z}\text{,}$$ I want to know a function $g:\mathbb{C}\to\mathbb{C}$, for which holds $$\left(f\circ ...
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2answers
26 views

Prove absolute convergence from alternants

He failed to get the show in its entirety in this series, one I could indicate how working with this kind of series? $$ \sum \limits^{\propto }_{n=1}\frac{(-1)}{n(\ln(n+1))^{2}} $$
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1answer
29 views

Formula for $q$-expansion of weight 2 modular forms

Is there a general formula for finding the $q$-expansion of weight 2 modular forms?
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1answer
19 views

Interval of convergence? (Relatively simple question)

What is the interval of convergence of the power series: $\dfrac{(-1)^{(n-1)}x^n}{n^3}$ I know it should be |x| < 1, but does that mean the interval of convergence is $(1,-1)$ or $(-1,1]$ or ...
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1answer
25 views

$z \cdot \cot(z)$ series

Let us consider an expansion $z \cot(z) = \sum_{n=0}^{\infty}{(-4)^{n} \cdot B_{2n} \cdot \frac{z^{2n}}{(2n)!}}$. How to prove the RHS? I see possible to come to the expansion $\pi \cot(\pi z) = ...
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2answers
27 views

Power Series of a Holomorphic Function determined by its Real Part and $f(0)$?

While looking at exercise sheets from last year, I encountered the following statement but wasn't able to prove it myself. Let $f: D_R(0) \rightarrow \mathbb{C}$ be holomorphic and $ f(z)= ...
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0answers
43 views

How to show that two probability generating functions are equal?

From Grimmett's Probability and Random Processes: Let $G_a(s) := \sum_0^\infty a_is^i$ where $a = \{a_i : i \geq 0\}$ is a real sequence. Uniqueness. If $G_a(s) = G_b(s)$ for $|s| < R'$ ...
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0answers
37 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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3answers
44 views

Power Series Coefficients

Find the sum of the coefficients of $x^{20}$ and $x^{21}$ in the power series expansion of $\frac 1{(1-x^3)^4}$. I don't know a lot on power series at the moment, and I was wondering how do I find ...
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1answer
31 views

generating function for a power sequence

The question is short: I don't understand how should I solve this. Problem wants the G(x) of this: 1,4,9,16,... I can solve this one but I cannot connect these two to each other: 1,2,3,4,...
131
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15answers
9k views

How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
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1answer
43 views

Complex Number question [Cauchy Integral/Series]

I'm going through the practice finals that my professor uploaded on his site, and I came across this question, and I have absolutely no one clue how to approach it and never seen anything like this on ...
2
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1answer
48 views

Find the general solution to 2y''+xy'+y = 0 in the form of a power series about the ordinary point x=0.

Question: Find the general solution to $2y''+xy'+y = 0$ in the form of a power series about the ordinary point $x=0$. My Working: Firstly $\space y = \sum\limits_{0}^{\infty} (A_n.(x-1)^n)$ ...
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2answers
32 views

How to do power series expansion

What is the coefficient of $x^{11}$ in the power series expansion of $\frac 1{1-x-x^4}$? How do I do power series expansions?
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2answers
54 views

Is this a power series?

Is the following a power series? $$\sum_{n=0}^\infty a_k \left( \frac{2x}{1+x^2} \right)^k \ , x \in (-1,1)$$ where $a_k$ is a bounded sequence. I was asked to show that this power series converges, ...
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3answers
446 views

Solution of $y''+xy=0$

The differential equation $y''+xy=0$ is given. Find the solution of the differential equation, using the power series method. That's what I have tried: We are looking for a solution of the form ...
2
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3answers
63 views

What is the technical difference between a formal and informal power series?

In my lecture notes the professor wrote that $$e^x = \Sigma \frac{x^k}{k!}$$ is a formal power series because we can plug in whatever we want in $x$ and both side will equate This is an obvious ...
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2answers
32 views

Condition for the convergence of a particular power series in $ℂ$

The problem is given as: Show that there exists no power series $f(z)=\sum_{n=0}^{\infty}C_nz^n$ such that:$f(z)=1$ for $z=\frac12,\frac13,\frac14,...$ and $f'(0)>0$ My approach so far: Let's ...
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2answers
83 views

How can I find the sum of this series?

The series is $$\sum\limits_{k=0}^\infty\frac{3k}{k!}x^{3k-1}$$ I already calculated $$\sum\limits_{k=0}^\infty\frac{x^{3k}}{k!}=e^{x^3}$$ So I tried to make the two look similar but I don't know ...
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2answers
46 views

Showing that $\sin'(x)=\cos(x)$

I want to show the "simple" relation: $$\sin' x=\cos x$$ by using power series. I know that: $$\sin x=\sum_{n=0}^{\infty}(-1)^n \frac{x^{2n+1}}{(2n+1)!}$$ $$\cos x=\sum_{n=0}^{\infty}(-1)^n ...
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0answers
27 views

Series identity for cotangent

How to prove that $x \cot(x) = 1 - 2 \sum_{n=0}^{\infty}{\frac{x^{2}}{(n \pi)^{2}-x^{2}}}$? First, it does not seem to be solvable, using considerations regarding Taylor series. The Fourier approach ...
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2answers
40 views

find interval of convergence for series

Is it right that the range of convergence is here $1 < x < 3$: $$\sum_{n= 1}^\infty \frac{e^n + e^{-n}}{n^2} (x-2)^n$$ Just like you do with the geometric series? Or what is this radius of ...
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0answers
41 views

How do I express each natural number as sum of serie?

I have many attempts to express each natural number as a sum of series which I meant not to take all convergents series that are giving us 1 as a result I want only how to let e.g : 1 defined ...
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0answers
8 views

domain of convergence of power series in many variables

If you have a power series in many variables with coefficients in $\mathbb{R}$ or $\mathbb{C}$ is there a result saying that the series is absolutely convergent in the interior of the set of points ...
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1answer
123 views

The power series $\sum_{n\geq 1} \frac{x^{n}}{n(2n-1)}$ with $2$nd Taylor polynomial and Taylor series. [Solved]

I have been a fool not noticing it earlier. Instead of deleting this thread I have chosen to put the short solutions of this problem. This thread is closed. Consider the series $$\sum_{n\geq 1} ...
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2answers
46 views

Finding $f^{(m)}(i)$ where $f(z) = (1 + (z - i)^2)^{-1}$ without differentiating.

I have a question and I'm not to sure how to approach it, so any kind of help will be awesome. I was given this question in the practice final, however there are no solutions/hints to this question, ...
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6answers
64 views

How can I expand $f(x)$ in powers of x?

$f(x)=\frac{1-x}{1+x}$. The closest thing I know to this would be $\sum_{k=0}^\infty x^k=\frac{1}{1-x}$ but I don't know how to use it to write $f(x)$
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4answers
97 views

Calculate the sum $\sum_{n=3}^{\infty}\frac{4n-3}{n^3-4n}$

$$\sum_{n=3}^{\infty}\frac{4n-3}{n^3-4n}$$ I think it is related to power series, because it is the topic, but I have no idea how to get there. Could you give a hint?
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3answers
45 views

Power series expansion of $f(z)=\frac{1}{3-z}$ about the point $4i$

I want to find the power series expansion of $f(z)=\frac{1}{3-z}$ about the point $4i$ and to find the radius of convergence, what does this take? Is this just a taylor series with $z=4i$ subbed in? ...
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1answer
35 views

Power series - interval of convergence

For $f(x) = \sum_{n=2}^{\infty} \frac{(x+1)^n}{n(n-1)}$ I have showed that $f'(x) = \sum_{n=1}^{\infty} \frac{(x+1)^n}{n}$ and that $f''(x)=\frac{-1}{x}$ at all points where f converges absolutely. ...
0
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1answer
11 views

How to determine the domain of convergence of the $k$-th derivative of power series

How to determine the domain of convergence of the $k$-th derivative of power series: $$\sum_{n=0}^{\infty}{(-1)^nz^n}$$ For all $k=1,2,...$ ¿Which functions does this series represents?
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1answer
44 views

Understanding series and their sums

Here's something that I can't wrap my head around while self-studying analysis. Is defining a function to be a series and defining a function to be the sum of a series considered to be two different ...
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1answer
45 views

A question about a polynomial

Suppose that $p$ is a real polynomial of degree $n$. Prove that for $|x|<1$, $$\sum\limits_{m=0}^\infty{p(m)x^m}=h((1-x)^{-1})$$ for some real polynomial $h$ of degree $n+1$ without the ...
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1answer
20 views

Power Series - differentiation and absolute convergence

I am having problems with the following exercise: Ex. 1. Let $f(x) = \sum_{n=1}^{\infty} \frac{(x-1)^n}{n}$ (i) Find the convergence interval. Here I let $f(x) = \sum_{n=1}^{\infty} ...
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1answer
57 views

Solution of differential equation - We find only one

I want to find all the solutions of the form $y(x)=x^m \sum_{n=0}^{\infty} a_n x^n, x>0 (m \in \mathbb{R})$ of the differential equation $x^2 y''+ xy'+x^2y=0$. I have tried the following: Since ...
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3answers
68 views

$f(x)=\sum_{n=0}^{\infty}a_n x^n$ and there exists a sequence $x_n\to 0$, such that $f(x_n)=0$, for all $n$. Then $f\equiv 0$.

I found this question really difficult for me, I don't even know how to start with it? Could you help me? I will appreciate that. Prove that if $f(x)=\sum_{n=0}^{\infty}a_n x^n$ (defined in ...
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4answers
672 views

Number raised to power of irrational number

What is the consequence of raising a number to the power of irrational number? Ex: $2^\pi , 5^\sqrt2$ Does this mathematically makes sense? (Are there any problems in physics world where we ...
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3answers
30 views

Find all the values of x, for which the series converges.

$\sum\limits_{n=1}^∞ (x^2/(x^2+4))^n$ I did try to use the ratio test and I ended up with $| x^2/(x^2+4)|<1$ I don't have any idea what to do after this, how do I solve for x?
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2answers
48 views

How does this manipulation of summations work?

I am reading some mathematics in which is the following algebraic manipulation. $$ \begin{align} \exp(x)\exp(y) & = \left(\sum_{n = 0}^\infty \frac{x^n}{n!}\right) \left(\sum_{m = 0}^\infty ...
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1answer
38 views

Differentiate this power series

I am working on a problem which involves the differentiation of a power series. I know that that the following holds. Let $R$ be the radius of convergence of the power series $\sum_{n = 0}^\infty ...
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0answers
42 views

Analytic continuation of ln(z) counterclockwise about the unit circle,

We write ln(z) as ln(1+z-1) = ln(1+(z-1)) to utilize the familiar expansion that is: (z-1) - (z-1)^2 / 2 + ... which converges for |z-1| < 1, i.e., we get convergence of ln(z) in an open Taylor ...
1
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1answer
49 views

Power series with coefficients in primary ideals

Let $P$ be a prime ideal in a commutative ring $R$ with unity such that an ideal $Q$ is $P$-primary and some power of $P$ is a subset of $Q$. I want to show that $\sqrt {Q[[x]]}=P[[x]]$. If a ...
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2answers
22 views

Determining Radius of Convergence of Power Series

Find the radius of convergence for the following power series: My workings: $$\lim_{n\rightarrow ∞}|\frac{(n+1)! (x-1)^{n+1}}{2^{n+1}(n+1)^{n+1}}\centerdot \frac{2^nn^n}{n!(x-1)^n}|$$ ...
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0answers
27 views

Proof that real power series is real analytic

I'm wondering if the following argument is correct. The proof in the book is longer and I don't understand it. Theorem. Suppose $f(x) = \sum_{n=0}^\infty a_n x^n$, where the series converges for $-R ...
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4answers
344 views

solution to differential equation from deriving power series

Find the solution of the differential equation $$y'= 2xy$$ statisfying $y(0)=1$, by assuming that it can be written as a power series of the form $$ y(x)=\sum_{n=0}^\infty a_nx^n.$$ Im advised to ...
2
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1answer
46 views

How do we derive the sum of $3^n$ and $2^n$

I know that $\quad\sum2^n = 2 (2^n-1)$ How can we derive this summation? And also how can we deduce the summation of $3^n$ from this ? I did observe this pattern : $$ \begin{align} n &= 1 ;\ ...
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0answers
13 views

Series $\sum_{d \geq 0} \sqrt{d} \cdot z^d$

As stated in the title, I'd like to compute the series $\sum_{d \geq 0} \sqrt{d} \cdot z^d$ where $z$ is a (small enough) complex number. More generally, for any real value $\alpha$, is there a ...
0
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1answer
31 views

Interval of convergence of power series?

If the power series is: $$ \sum\limits_{n=1}^{\infty}\frac{x^n}{\sqrt{n+1}} $$ and I've found the interval to be $$ -1 < x < 1 $$ then would the answer $$ (-1, 1) $$ work? some other options ...