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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2
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2answers
35 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?
3
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2answers
55 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, ...
2
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3answers
68 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 ...
0
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2answers
35 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 ...
2
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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 ...
0
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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 ...
3
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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 ...
2
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1answer
57 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 $\space y = \sum\limits_{0}^{\infty} (A_n.(x-1)^n)$ ...
2
votes
2answers
41 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 ...
0
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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
10 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 ...
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, ...
0
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6answers
66 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 $f(x)$
1
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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?
1
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3answers
46 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? ...
0
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1answer
12 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?
1
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1answer
38 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. ...
5
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1answer
46 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 ...
0
votes
1answer
21 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} ...
1
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1answer
59 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 ...
2
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3answers
31 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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3answers
69 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 ...
2
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2answers
50 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 ...
1
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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 ...
0
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1answer
42 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
50 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 ...
0
votes
2answers
22 views

Determining Radius of Convergence of Power Series

Find the radius of convergence for the following power series: My workings: $$\lim_{n\rightarrow ∞}|\frac{(n+1)! (x-1)^{n+1}}{2^{n+1}(n+1)^{n+1}}\centerdot \frac{2^nn^n}{n!(x-1)^n}|$$ ...
0
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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 ...
6
votes
4answers
381 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
votes
1answer
47 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 ;\ ...
2
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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
votes
1answer
126 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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0answers
36 views

Weird question about interval of convergence

The question is: if $$ f(x) = \sum \limits_{n=0}^\infty x^n$$ determine the interval of convergence for the power series representation of $$\int_0^x f(t) \, dt$$ That integral threw me off.
1
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1answer
33 views

Power series representation?

The function to represent as a power series is: $$ \frac {10} {(x-10)^2} $$ Any help is, as always, appreciated.
0
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1answer
32 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 ...
2
votes
3answers
38 views

(Simple question) Radius of covergence of power series?

For the power series: $$ \sum\limits_{n=1}^{\infty}\frac{(x-1)^n}{2^n} $$ Would radius of convergence be $$ x = 1 $$ ?
0
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1answer
38 views

Determine the radius of convergence of the power series

Determine the radius of convergence of the power series $\sum \limits _{n=4} ^\infty \frac {2n+4} {4^{n+5}} (x-8)^{4n+1}$. I tried the ratio test to find where $\frac {a_n} {a_{n+1}} < 1$ but I ...
2
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1answer
46 views

What is the center of power series?

The power series is: $$ \sum\limits_{n=1}^{\infty}\frac{(x+4)^n}{n+1} $$ Any help appreciated!
2
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0answers
43 views

Does convergence of power series on radius of convergence imply absolute convergence?

Let $R$ be radius of convergence of power seires $\displaystyle\sum_{k}a_kz^k$. If the power series converges for all $|z|=R$, can we say that it converges absolutely on the radius of convergence? I ...
0
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1answer
24 views

Show that if $f(z)=\frac{\operatorname{Log}z}{z-1}$ when $z\neq 1$ and $f(1)=1$, then $f$ is analytic throughout the domain.

$\operatorname{Log}z=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}(z-1)^n \; (|z-1|\lt 1).$ Use this fact to show that if $$f(z)=\frac{\operatorname{Log}z}{z-1} \; \text{when} z\neq 1$$ and $f(1)=1$, ...
2
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1answer
56 views

Convergence of complex power series $z^{n!}$ at boundary

I'm revising for an exam at the moment and I'm struggling with part of a question. I'm asked to find the radius of convergence of the series $\sum_{n=0}^{\infty }z^{n!}$ and then find where it ...
0
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1answer
39 views

With the aid of series, show that if $f(z)=\frac{\operatorname{cos}z}{z^2-(\pi/2)^2}$, then $f$ is an entire function.

Prove that if $$f(z)= \begin{cases} \frac{\operatorname{cos}z}{z^2-(\pi/2)^2}, & \text{when} \; z\neq \pm \pi/2, \\ -\frac{1}{\pi}, & \text{when} \; z=\pm \pi/2, \end{cases} $$ then $f$ is ...
3
votes
1answer
92 views

Is $\mathbb{Q}(\pi) \cong \mathbb{Q}[[x]]/ \langle \sin(x) \rangle$?

If we let $\mathbb{Q}[[x]]$ be the set of all power series with rational coefficients then can we say that $\mathbb{Q}(\pi) \cong \mathbb{Q}[[x]]/ \langle \sin(x) \rangle$?
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votes
1answer
35 views

“Sum of power” for prime numbers

I use Euler–Maclaurin formula, Faulhaber's formula and Bernoulli polynomials for "sum of powers" for this type $\sum_{t=1}^nt^k$. but I don't know to find compact form when sum is taken from first ...
2
votes
1answer
55 views

Show that the binomial series satisfies: $(1+x)f'(x)=\alpha f(x)$

If $f(x) = \sum_{n=0}^{\infty}{\alpha\choose n}x^n$, show (without assuming the results of the binomial theorem) that $$(1+x)f'(x)=\alpha f(x)$$ for $|x|<1$ I've already shown that the sum ...
0
votes
1answer
45 views

Power Series Expansion

How can I find the Maclaurin series for $f(x)=e^x$/$(1-x^2)$? I have tried expanding it out but I am having trouble with the algebra of it.
0
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0answers
46 views

Computing an exponential generating function from the first few terms

The current question is related to this one, and this other one. I have a number sequence, and I want to find generating ...
4
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0answers
40 views

Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, ...
3
votes
1answer
93 views

Power series as fractions

This is what I did: \begin{equation*} (x^3-x^6)x^6[x+x^2+x^3+..], \\ \frac{(x^3-x^6)x^6}{1-x}. \end{equation*} What mistake did I make? And, How to solve this: $1+3x^2+9x^4+27x^6+...+3^{157}x^{314}$ ...
4
votes
1answer
85 views

How many $s,t,u$ satisfy: $s +2t+3u +\ldots = n$?

Given $n\in \mathbb{N}^+$, what is the possible number of combinations $s,t,u,\ldots\in\mathbb{N}$, such that: $$s +2t+3u +\ldots = n\quad?$$ Additionally, is there an efficient way to find ...