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Questions tagged [power-series]

Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n (x-c)^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex.

3
votes
3answers
585 views

Is there a name for this series?

I am doing a presentation on nonlinear optics and I ran into a paper that uses a series to describe a wave equation of a system. The paper will be linked below and the series is: $\sum_{n=1}^{\infty}\...
0
votes
1answer
23 views

True or False: If $x\notin \mathbb{Q}$ then $\sum_{m\geq 0} mx^{m-1}\notin \mathbb{Q}$

True or False: If $x\notin \mathbb{Q}$ then $\sum_{m\geq 0} mx^{m-1}\notin \mathbb{Q}, $ where $|x|<1.$ So I considered the contra-positive of the above statement: If $\sum_{m\geq 0} mx^{m-1}\in \...
-2
votes
2answers
34 views

For $|x|<1/7$ find $\sum_{n = 0}^ {\infty} n^27^nx^n$.

I have figured out three of them but I am stuck on the last one, can anyone help out? $$ \begin{split} \sum_{n = 0}^ {\infty}7^nx^n &= \frac1{1-x} \\ \sum_{n = 0}^ {\infty}n7^nx^{n-1} &= \...
-2
votes
0answers
15 views

power distribution network [on hold]

I work on the power distribution network. To test the convex optimization algorithm, I need to simulate the software on a IEE-123 standard bus. Which software is suitable for this job? Is MATLAB ...
3
votes
1answer
43 views

$(a_n) $ is a sequence of positive real numbers. The series $\sum a_n$ will converge if

$(a_n) $ is a sequence of positive real numbers. The series $\sum a_n$ will converge if (a) $\sum a_n^2$ converges. (b)$\sum \frac{a_n}{2^n}$ converges (c)$\sum \frac{a_{n+1}}{a_n}$ coverges (d)$\...
2
votes
1answer
25 views

Let $(a_n),(b_n) $ be sequences of positive real numbers such that $na_n<b_n<n^2a_n$

Let $(a_n),(b_n) $ be sequences of positive real numbers such that $na_n<b_n<n^2a_n$ for all $n\geq2$. If the radius of convergence of $\sum a_n x^n$ is $4$, then $\sum b_n x^n$ A)converges ...
-1
votes
1answer
32 views

Coming up with an example of a subspace that is not closed.

I am trying to include the fact that $\lim_{N\to\infty} \sum_{n=0}^{N} x^n=\frac{1}{1-x}$ for $|x|<1$. Perhaps I could use $C[0,1]$ as a subspace of $C(\mathbb{R})$(or some bigger space?), and ...
1
vote
1answer
25 views

Convergence of a series of a sequence of powers

Let $\rho\in]0,1[$ and $(u_n)_{n\in\mathbb{N}}$ a sequence of positive integers such that $\lim_{n\to\infty}u_n=\infty$. Consider the following series: $$\sum_{n\in\mathbb{N}}^\infty \rho^{u_n}$$ Q : ...
8
votes
1answer
330 views

A power series with decreasing positive coefficients has no zeroes in the disk

Let $\sum_{n=0}^\infty a_n z^n$ be a formal complex power series, with $a_n$ strictly decreasing to 1 as $n\to \infty$. It is easy to see that the radius of convergence is 1, so that this power series ...
0
votes
2answers
32 views

Find the convergence radius for this power series

The Problem: Find the convergence radius of $\sum_{n=0}^{\infty} \frac{n}{5^{n-1}} z^{\frac{(n)(n+1)}{2}}$ My attempts to find a solution I apply either the ratio test and end up with this ...
2
votes
1answer
36 views

Series solution of $(1-x)^3y’’-6x^2y-6xy=0$

For the differential equation $(1-x)^3y’’-6x^2y-6xy=0$, I considered a series solution about the ordinary point $x=0$. Using a series solution, I got a very nice recurrence relation of simply $$a_{n+...
2
votes
0answers
50 views
+50

Completion of the local ring at a point on arithmetic surfaces.

Let $K$ be a number field and consider a arithmetic surface $X\to B=\operatorname{Spec} O_K$, i.e. $X$ is integral, regular, flat, proper over $O_K$ and it has dimension $2$. Now pick a closed point $...
0
votes
0answers
19 views

How to express a series expansion into a power law form?

Suppose I have a series like $P\sim A+\epsilon B\,lnM+O(\epsilon ^{2})$ and I want to express it into a power law form. The answer is $P\sim AM^{\epsilon B/A}+O(\epsilon ^{2})$. Another example. ...
2
votes
1answer
26 views

Contradiction in radii of convergence? Where is my error?

I'm working through Baby Rudin and I came across what seems to me to be a contradiction, but it could be an error on my part. It has to do with radii of convergence of power series. First, let $\{...
2
votes
4answers
62 views

How to “directly” obtain Maclaurin series of $\exp(x-1+\sqrt{x^2+1})$

Consider the Taylor expansion centered around $x_0 = 0$ for $$f(x) = e^{x-1 + \sqrt{x^2+1}} ~,\qquad x\in\mathbb{R}$$ The goal is to arrive at $\sum c_k x^k$ by hand, and I wonder if there is an ...
5
votes
0answers
64 views

Is the sum of series $\sum_{n=0}^{\infty} \lfloor n\pi \rfloor x^n$ a rational function?

I was reading this paper by L.J. Mordell (original paper by Morris Newman). I am trying to apply first theorem from the paper with $f(x)=x$ and $k=\pi$. So the series $\displaystyle \sum_{n=0}^{\infty}...
0
votes
0answers
28 views

Cohen Structure Theorem Hensel's Lemma

I have a question about a comment occurred in the following former thread from MO: https://mathoverflow.net/questions/309609/cohen-structure-theorem-with-explicit-equations With the same setting as ...
9
votes
2answers
276 views

Ramanujan's Master Theorem relation to Analytic Continuation

$\DeclareMathOperator{Re}{Re}$ To provide some background, this is a question based on establishing the identity $$\int_0^\infty \frac{v^{s-1}}{1+v}\,dv=\frac{\pi}{\sin \pi s},\qquad 0<\Re s<1$$...
0
votes
2answers
29 views

Find rational representation of a power series

I need to find a rational function $\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials, which value is the same as $\sum_{n = 0}^\infty (2n+1)z^{2n}$ on its convergence domain. I found $\rho=...
2
votes
0answers
47 views

If $f(x)=\sum\limits_{n=0}^\infty a_n x^n$ is even and continous, then $a_n=0$ for odd n, and if f is an odd function, then $a_n=0$ for even n.

I need help with this problem: Show that If $f(x)=\sum\limits_{n=0}^\infty a_n x^n$ is an even and continous function, then $a_n=0$ for odd $n$, and if f is an odd function, then $a_n=0$ for even $...
0
votes
1answer
36 views

Property of alternating sign series

I have an alternating series $\sum_{i=0}^{\infty}(-1)^{i+1}a_ix^i$, with $a_i\geq 0$ and the series is easy to check to converge for any $x>0$. I numerically checked that this sum is negative for ...
-4
votes
1answer
35 views

Find the power series representation of $xe^x$ [closed]

I'm having such difficulty in trying to find the power series representation for this function $$xe^x$$. I know the power series representation for $e^x$, which is $$e^x=\sum_{n=0}^\infty \frac {x^n}{...
1
vote
1answer
30 views

Proving “$\forall_{\epsilon>0} \exists_{N \in \mathbb{N}} \forall_{n \geq N}: x^{n} < \epsilon$” for $0<x<1$.

I'm trying to prove a rather simple analysis statement but I think I'm overseeing something. For $0<x<1$ I need to prove $$\forall_{\epsilon>0} \exists_{N \in \mathbb{N}} \forall_{n \geq N}...
0
votes
0answers
17 views

Convergence of Power Series and Power Series Solutions of ODE

A function which has a convergent power series expansion about a point is called analytic at that point. A function may not be analytic at some points but analytic every where else. This means that ...
0
votes
1answer
41 views

Find $f^{(k)}(0)$ of $f(x)=\frac{\sin x}{x}$ for $x\neq 0$ and $f(0)=1$.

I'm stuck with this problem: If $f(x)=\frac{\sin x}{x}$ for $x\neq 0$ and $f(0)=1$, find $f^{(k)}(0)$. Hint: Find the power series of f. I tried to solve it by writing down the Taylor series of $$\...
1
vote
1answer
68 views

Show that $e^{-1/x^2}$ is not analytic around $x=0$.

I have been working on the following question. Define a function \begin{align*} f(x)= \begin{cases} e^{-1/x^2}&\text{ for }x>0,\\ 0&\text{ for }x=0 \end{cases} \end{align*} Prove that $f$ ...
2
votes
1answer
23 views

Radius of convergence of $f(x)=\arcsin(x)$.

I am working out the series representation for the $\arcsin(x)$ function and its radius of convergence, I'm just not sure if my calculations are correct. I used the generalized binomial formula to ...
0
votes
1answer
31 views

Uniform convergence of series of functions given convergence of coefficients

Prove that if $\sum{a_n}$ converges then the series $\sum{a_nx^n}$ converges uniformly on [0,1] I believe that I must use the Weierstrass M-test to show this convergence, but this requires that the ...
1
vote
1answer
22 views

Show that $ \int_0^xe^{x-t}f(t)dt=\sum_{n=0}^\infty( f^{(n)}(0)(e^x-\sum_{i=0}^n \frac{x^i}{i!}))$ ,$-R\lt x\lt R$.

For a function $f$ that has a power series representation centered at the origin, with radius R>0, show that : $F(x)=\int_0^xe^{x-t}f(t)dt=\sum_{n=0}^\infty( f^{(n)}(0)(e^x-\sum_{i=0}^n \frac{x^i}{i!})...
2
votes
2answers
81 views

How to find $\int \sum_{n=0}^\infty \frac {(\ln x)^n}{x^n.n!}dx $?

I know how to find the integral of a normal power series like the power series of $ \sin (x) $, but have no idea how to integrate the $( \ln x )^n $ term inside the summation.
1
vote
1answer
56 views

Series Solution to the ODE $y''+2y'+y=0$

here's where I'm at, The ode we are trying to solve is, $$y''+2y'+y=0$$ I know this solution is of the form: $$\sum_{n=0}^{\infty}a_nx^n $$ And from that we get the following $$\sum_{n=0}^{\infty}(n)(...
1
vote
1answer
155 views

How would we find $ \int \sqrt[x]{x}$ or $\int \left( x^{x^{x^{.^{… \infty}}}} \right) dx$?

I know that the antiderivate of $$x^{\frac{1}{x}}$$ cannot be expressed in the form of elementary functions, but even so how will we express it in the form of a power series ? Inspiration of the ...
1
vote
2answers
38 views

Power series solution to $y’ + 2xy = 0$

So I'm stuck. I'm pretty sure the constant $c_1$ equals zero, which makes the equation easy to solve by using the identity principle. But how do I show that $c_1=0$.
4
votes
3answers
142 views

What is the series expansion of $ \sqrt[x] x $?

What is the series expansion of $ \sqrt[x] x $ ? I want to find it because I want to find $ \int \sqrt[x] x dx$ but believe that the integral cannot be expressed in the form of elementary functions.
2
votes
1answer
48 views

How to find the second solution to an ODE of the form $xy''+y'-xy=0$?

I am learning about the Frobenius method to solving ordinary differential equations, but I seem to have some difficulty with finding the second linearly independent solution to the above equation (...
1
vote
1answer
22 views

Uniform convergence of $f_n (x)=x^{2} - \frac{x}{n}$ on $[0, 1]$.

I need help with this problem: Consider the sequences of functions $(f_n)$ defined by: $$f_n (x)=x^{2} - \frac{x}{n}$$. Show that $(f_n)$ converges uniformly to $f$ on$[0, 1]$. I've already found ...
0
votes
1answer
41 views

Radius of convergence of $\sum_{n\ge 1}\frac{z^n}{n}$ [duplicate]

In one exercise it is asked to find the radius of convergence of $$\sum\limits_{n\ge 1}\dfrac{z^n}{n};$$ then it is asked to find two values $z_1,z_2\in U=\bigg\{z\in\mathbb{C},|z|=1\bigg\}$ such ...
1
vote
1answer
35 views

Proving formal power series $A(x) = \sum_{n = 0}^{\infty} a_nx^n$ with $A'(x) := \sum_{n=0}^{\infty}(n+1)a_{n+1}x^n$

To the formal power series $A(x) = \sum_{n = 0}^{\infty} a_nx^n$ with the multiplicative inverse $A^{-1}(x)$ we define $A'(x) := \sum_{n=0}^{\infty}(n+1)a_{n+1}x^n$ How can one prove $$(A^{-1})'(x) =...
1
vote
0answers
20 views

On the Newton Polygon for $p-$adic Power series

I'm studyng a Book about $p-$adic numbers, and I have troubles with a "degenerate" case of a Newton polygon. Let $f(X)=\sum a_{i}X^{i}\in\mathbb{Q}_{p}[\![X]\!]$, we define the Newton poligon of $f$ ...
2
votes
0answers
49 views

Radius of convergence of series $\sum a_n x^{n^2}$ such that $a_0=1$ and $a_n=3^{-n}a_{n-1}$ for all $n\in\mathbb N$

My aim is to find the radius of convergence of the series $\sum a_n x^{n^2}$ such that $a_0=1$ and $a_n=3^{-n}a_{n-1}$ for all $n\in\mathbb N$. I will use the Ratio test, let $c_n=a_nx^{n^2}$ $$\lim ...
0
votes
2answers
42 views

Calculating $\sum_{n=0}^\infty\frac{(n+1)(n+2)}{2}x^n$ when $|x|<1$

How do I calculate the sum $$\sum_{n=0}^\infty\frac{(n+1)(n+2)}{2}x^n$$ I know this sum will be finite for $|x|<1$ and will diverge for other values of $x$. Since for other sums it was common to ...
0
votes
3answers
43 views

If a complex power series diverges at a point $z=z_0$ then it diverges for all $z$ satisfying $|z|>|z_0|$

If a power series $\sum a_nz^n$ is divergent at $z=z_0$, then how do we show that the power series is divergent for all $z$ satisfying $|z|>|z_0|$.
4
votes
1answer
88 views

Radius of convergence of $\sum_{n=0}^{\infty}(2^n)x^{n^2}$

Radius of convergence of $\sum_{n=0}^{\infty}2^nx^{n^2}$ I will use the Root test, let $c_n = 2^nx^{n^2}$ $\lim_{n\rightarrow \infty} c_{n}^{1/n} = \lim_{n\rightarrow \infty} 2x^{n} = 0 $ if $-1<...
0
votes
2answers
55 views

Radius of convergence of power series $\sum_{n=0}^{\infty} n!x^{n^2}$

Radius of convergence of power series $\sum_{n=0}^{\infty} n!x^{n^2}$ $\sum_{n=0}^{\infty} n!x^{n^2} = 1 + x + 2x^4 + 6x^9\ldots$ Comparing this with $\sum_{n=0}^{\infty} a_nx^n=$ $a_n= n! $ or $0$ ...
0
votes
2answers
55 views

Radius of convergence of $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}z^{n(n+1)}$

I'm interested in finding the radius of convergence of the power series $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}z^{n(n+1)}$. My first step was to rewrite this series into the standard power series form ...
2
votes
4answers
59 views

Find the power series expansion of $f(x)= \frac{e^x - 1}{x}$

Prove that the following function is analytical in 0 and find its power series centered in 0 $$f(x)= \frac{e^x - 1}{x}, f(0)=1$$ I'm trying to write $f$ as some kind of combination of function ...
1
vote
1answer
45 views

What's the power series solution of $y' = x^2\,y$

My attempted solution I'm not sure how to factor out $x$. In the other problems I've solved, both series terms have the same power of $x$. but in this problem one series contains $x^{n+2}$ and the ...
1
vote
2answers
39 views

Solving $\sum_{n=1}^{\infty}(-1)^{n+1}(\frac{1}{n})$ using Abel's Theorem

I am looking for some help with an real analysis problem that I have. Problem: Find the sum of the series $\sum_{n=1}^{\infty}(-1)^{n+1}(\frac{1}{n})=1-(1/2)+(1/3)-(1/4)+\ldots $ What I have so far:...
1
vote
1answer
91 views

Solving $y'' - 2xy' - 2y = 0$ using power series.

$$(1) : y'' - 2xy' - 2y = 0 $$ Determine the power series solutions to the equation (1). Let $y$ be a a solution to the equation $(1)$, defined on $\mathbb{R}$ and from $C^2$, prove that ...
2
votes
3answers
129 views

Calculation of $\lim\limits_{n\to \infty }\frac {(2n)!}{(2^n(n)!)^2} $

I've had a hard time computing the limit $\lim\limits_{n\to \infty }\frac {(2n)!}{(2^n(n)!)^2} $ either by bounding it or by simplifying it. I would appreciate some help. (P.S. I came across this ...