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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1answer
52 views

Taylor series and radius of convergence: $\sqrt{x}$ with centre $x = 16$?

I've been struggling with this question for a while now and getting nowhere with it. Could someone please help me out? Assuming that the function has a power series expansion about the given point, ...
0
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1answer
36 views

Standard technique for fiddling with power series

I will try a standard technique for fiddling with power series. If $g(t) =\sqrt{f(t)} $, then, differentiating, $g'(t) =\frac {f'(t)}{2\sqrt{f(t)}} =\frac {f'(t)}{2g(t)} $ so $2g'(t)g(t) = f'(t) $. ...
2
votes
1answer
41 views

Convergence of power series dependent on parameter

I want to prove that $$\sum_{n=1}^{\infty}{\frac{x}{n^{\alpha}(1+nx^2)}}$$ converges for each $x\in\mathbb{R}$ whenever $\alpha>\frac{1}{2}$. How can I prove this. Which test I have to use?
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1answer
57 views

How to solve $xy''+2y'+\lambda^2 \, xy=0$ with the power series method?

Find all functions $y(x)$ which can be expressed as a convergent power series $y(x)=\sum_{n=0}^{\infty} a_n x^n$ and which satisfy the following differential equation: $$xy''+2y'+\lambda^2 \, xy=0$$
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1answer
18 views

Discrete mathematics power series

I've hit a wall with this problem. So, i have been given: $A(x)=a0+a1*x+a2*x^2...ak*x^n$ and $B(x)=1/(1-x)*A(x)$ How do i show that $[x^n]*B(x)=sum(ak, k=0 .. n)$. I cant find a way through how ...
0
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1answer
31 views

radius of convergence of integral of power series

Suppose $\sum \limits_{n=0}^{\infty} a_n x^n$ has radius of convergence R. What is the radius of convergence of $\sum \limits_{n=0}^{\infty} \frac{a_n x^{n+1}}{n+1}$? How do I solve this without ...
1
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1answer
20 views

Find a power series representation for the function. (Assume $a > 0$.)

I'm down to my last attempt (my teacher allows $5$ tries per question)! Thank you!!
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1answer
22 views

The radius of convergence of a power series.

If I have a power series $$\sum_{j = 1}^{\infty}a_jx^{2j+1} = x\sum_{j = 1}^{\infty}a_jx^{2j} $$ Given that I have the radius of convergence $R$ of $$\sum_{j = 1}^{\infty}a_jz^{j}$$ where $z = ...
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0answers
50 views

Prove that the field of Puiseux series over $\mathbb C$ is algebraically closed

Denote by $K=\mathbb{C}((z))$ the fraction field of $\mathbb{C}[[z]]$. Define an embedding of $K$ onto itself taking $a(z)$ to $a(z^n)$ $\forall n$. The target is $\mathbb{C}((z^{1/n}))$. Define the ...
1
vote
1answer
37 views

Find a power series representation (centered at x = 0) and determine the radius and interval of convergence

For the following function: $f(x) = (x/(2-x))^3$ How do I find a power series representation (centered at x = 0) and determine the radius and interval of convergence? I managed to simplify the ...
1
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1answer
65 views

Calculating $\sum_{n=1}^\infty {\frac{nx^n}{4n^2-1}}$ [closed]

I would appreciate any help calculating the series. And determine where does the series converge uniformly. $$\sum_{n=1}^\infty {\frac{nx^n}{4n^2-1}} $$
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1answer
49 views

Inverse for $1-zb(z)$

I need to find the inverse of $1-zb(z)$ with $b(z)=\sum_{n=0}^{\infty}b_nz^n$. I have tried several approaches where I among other things have tried using the methods in my calculus book but nothing ...
0
votes
2answers
28 views

Power series expansion

I am trying to solve a equation which I have already solved using oDE but I want to solve it using a power series expansion but how do I express y as a power series? Equation is as below $$ (1+x) ...
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0answers
24 views

Reference request: attempt to give credit for power series result.

I've been informed that the result below is known by different names in different fields and I'm simply looking for the best person to credit. To my knowledge it originates with Kemp & Kemp ...
4
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5answers
276 views

Power series summation [closed]

Trying to find the sum of the following infinite series: $$ \displaystyle\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{(2n-1)3^{n-1}}$$ Any ideas on how to find this sum?
1
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1answer
41 views

Different series representation for the same function

Ok, we know that $\frac{1}{1-x}=1+x+x^2+x^3+\cdots$. Now if we want to have a series representation of $y=\frac{1}{2-x}$ then there are two approaches: First ...
0
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1answer
28 views

Power series for non elementary functions

Since the function $f(x)=e^{-x^2}$ cannot be integrated using elementary functions, how could one find a power series for $F$, where $F$ is an elementary function such that $F'(x)=e^{-x^2}$?
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1answer
27 views

If a series is not a power series, is the notion of radius of convergence still legit?

This notion is defined over complex power series. But what about complex series not in a "power" form? Is the convergence region always form a circle centered at origin in the complex plane?
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0answers
23 views

Frobenius Method, Power series

Im trying to self-learn method of frobenius, and I would like to ask if someone can explain to me how can we solve the following DE about $x=0$ using this method. $$ x^2y'' ...
2
votes
1answer
42 views

Power Series $x^2y''+y=0$

Actually, I tried to solve (x^2)y''+y=0 power series, I cannot get the general soultion or the relation at least $(x^2)y''+y=0 $ $ \sum_{n=2}^\infty c_n n(n-1) x^n + \sum_{n=0}^\infty c_n x^n$ $ ...
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2answers
39 views

General Solution to Differential Equation

Find the general solution of a differential equation $\frac d{dr}[r^2 \frac{dR}{dr}]-l(l+1)R=0$. (Hint: Assume an infinite series $R(r)=\sum_{n= -\infty}^{+\infty} a_n r^n$ as the solution) I don't ...
0
votes
2answers
44 views

Condition for convergence of infinite sum $\sum_{k=1}^{\infty}\frac{x^k}{k} $

Consider the following: $Q= \displaystyle\sum_{k=1}^{\infty}\frac{x^k}{k} $. What condition is required for $x$ so that $Q$ becomes convergent?
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2answers
31 views

What is the proper adjective/adverb for a power function?

I have a function where space grows as a power of time: $x= at^2$. In my report, I've been using the adjective 'exponential' or adverb 'exponentially' to describe the expansion with time. However, ...
5
votes
1answer
79 views

Evaluate the improper integral $ \int_0^1 \frac{\ln(1+x)}{x}\,dx $

I am trying to evaluate $$ \int_0^1 \frac{\ln(1+x)}{x}\,dx $$ I started by using the Taylor series for $\ln (1+x)$ $$\begin{align*} \int_0^1 \frac{\ln(1+x)}{x}\,dx &= ...
2
votes
5answers
61 views

Series expansion for $x$, when $x$ is small

Suppose that we are given the series expansion of $y$ in terms of $x$, where $|x|\ll 1$. For example, consider $$y=x+x^2+x^3+\cdots\qquad\qquad\qquad (1).$$ From this I would like to derive the series ...
1
vote
1answer
7 views

Writing power series for $AR(2)$ model polynomials

So I have found the following problem in my textbook without solutions, which presents the $AR(2)$ process defined by $$X_{t} = 0.5X_{t-1} + 0.25 X_{t-2} + Z_{t}$$ I am asked what the polynomial ...
0
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0answers
17 views

If a locally convergent power series factorises in $\Bbb{C}[[x,y]]$, then each factor is locally convergent

I've just started reading through Miranda's "Algebraic Curves and Riemann Surfaces", and there's one small bit I can't seem to work out: it seems like it should be easy, and indeed the author says ...
0
votes
0answers
45 views

How to solve $xy''+y'+m²y = 0$?

What I had in mind was $y=\sum_{n=0}^\infty a_k x^k$ $y'=\sum_{n=1}^\infty ka_k x^{k-1}$ $y''=\sum_{n=2}^\infty k(k-1)a_k x^{k-2}$ $\sum_{n=2}^\infty k(k-1)a_k x^{k-1} + \sum_{n=1}^\infty ka_k ...
2
votes
2answers
45 views

If a power series has positive radius of convergence and is non-constant within radius of convergence , then is all the zeroes of the series isolated?

Let $f(x)=\sum_{n=0}^\infty a_n x^n$ be a real power series with positive radius of convergence $R$ (including $R=+\infty$) , then we know that $f$ is continuous in $(-R,R)$ , so the zero set of $f$ ...
0
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0answers
51 views

Simplification of ODE using Power Series methods to find a solution

I have to substitute the power series $$y(x,r)=\sum_{n=0}^{\infty}c_n(r)x^{n+r}$$ into the ODE $L[y]=x^2y''+xp(x)y'+q(x)y=0$ which I need to simplify to $L[y(x,r)]=x^r(r-r_1)^2$ I've got as far ...
2
votes
2answers
43 views

evaluating limit using binomial series

I am trying to evaluate the following limit by using binomial series $(1+x)^{1\over 2}=\left ({1/2}\over n\right) x^n$ $$\lim_{x\to 0}{{(1+x)^{1\over 2}-1-{1\over 2}x +{1\over 8}x^2}\over ...
0
votes
1answer
20 views

Limit of analytical function

Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of analytical functions in $U \subset \mathbb{R}$ ($U$ open), ie. $f_n(x) = \sum_{k=0}^\infty c_{nk}(x_0) (x-x_0)^k$ for all $x_0 \in U$ and $x \in U$ with ...
3
votes
1answer
27 views

Power series differentiability at endpoints

I have the following problem: Find domain $I$ of the function defined by $f(x)=\sum\limits_{n=1}^{\infty}(3^{\frac{1}{n^2}}-1)x^n.$ Investigate differentiability of $f(x)$ in the interior of ...
0
votes
1answer
62 views

First five terms of power series

I want to find first $5$ terms for power series of $\frac{1 - x + 2 x^2}{1 - 5 x^3 + 3 x^6}$. $a_0 = \frac{1 - x + 2 x^2}{1 - 5 x^3 + 3 x^6} (0)$ = 1 $a_1 = \frac{\partial \frac{1 - x + 2 x^2}{1 - 5 ...
1
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2answers
90 views

Radius of convergence $\,\displaystyle\sum_{k = 0}^{\infty}\frac{ k^{2 k + 5} \ln^{10} k \ln \ln k}{\left(k!\right)^2}\, x^k$

I want to find the radius of covergence of $$\sum_{k = 0}^{\infty}\frac{ k^{2 k + 5} \ln^{10} k \ln \ln k}{\left(k!\right)^2} \,x^k$$ I know formulae ...
8
votes
1answer
122 views

Euler Transform elementary Proof

In this webpage Computing the Digits in π there is a proof of the Euler Transform (page 22). The proof there relies on measure theory and Lebesgue integration, I haven't studied that yet. In page ...
1
vote
1answer
51 views

Series expansion of natural log $\ln(1+1/x)$

I'm trying to get the series expansion of: $$f(x)=\ln\Big(1+\frac{1}{x}\Big)=\frac{1}{x}-\frac{1}{2x^2}+...$$ The series expansion of $\ln(1+x)$ around zero: ...
2
votes
0answers
40 views

Is it valid to make this integral and summation switch?

Is this ok to $$\sum_{i=1}^{\infty} a_i \int_a^b x^2 dx$$ $$ \lim_{n \to \infty} \sum_{i=1}^{n} a_i \int_a^b x^2 dx$$ $$\lim_{n \to \infty}\int_a^b\sum_{i=1}^{n} a_i x^2 dx$$ since rewriting the ...
3
votes
1answer
34 views

Frobenius Method Indicial Equation

I need to verify that the indicial equation only has one root. $xy''+(1-x)y'+\frac{1}{2}y=0$ Attempt: $y=\sum\limits_{m=0}^\infty {a_mx}^{m+r}$ $y'=\sum\limits_{m=0}^\infty {(m+r)}{a_mx}^{m+r-1}$ ...
1
vote
1answer
26 views

radius of convergence of function that is less than a polynomial

Find the power series and radius of convergence of $f(x) = (1+x^2)sin\left(x\right)$. I have found the power series. but for the radius of convergence can I say: $f(x)\leq 1+x^2 \implies$ the ...
1
vote
1answer
35 views

Power Series Expansion of Moment Generating Function

Suppose that X ∼ Γ(2, λ). Use the power series expansion for $M_X (θ)$ to determine $E(X^4)$. Working: I know that $M_X(\theta)=(\frac{\lambda}{\lambda - \theta})^2 = \frac{\lambda ^2}{(\lambda - ...
2
votes
1answer
44 views

Sum of series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{(2n-1)3^{n-1}}$

How do I work out the sum of $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{(2n-1)3^{n-1}}$$ I have $\sum_{n=1}^{\infty} \frac{(\frac{-1}{3})^{n-1}}{(2n-1)}$ I let $x=\frac{-1}{3}$. So ...
1
vote
1answer
30 views

How is this move in swapping integral with summation justified?

Here's my justification for swapping an integral symbol (with finite limits) with an infinite summation: EDIT after speaking with an MSE contributor, I think it is now apparent that the justification ...
2
votes
2answers
125 views

My proof that this series sums to log 2, [duplicate]

EDIT: Please see my attempt below at a proof to show the answer is $log2$ Let $$\alpha = \lim_{n \to \infty} \sum_{j=1}^n \frac {(-1)^{j+1}}{j}$$ Part (1): Show that $\alpha$ exists. Part (2): Show ...
0
votes
0answers
28 views

Inequalities involving Maclaurin Series of $1-e^{-x}$

Suppose you are given with the first three terms of the Maclaurin Series expansion of the function $f(x)=1-e^{-x}$ Then show, with reasoning, that $f(x)>x-x^2/2$ for $0<x<2$. Now, I ...
1
vote
3answers
45 views

$S(x)=\frac{x^4}{2\cdot 4}+\frac{x^6}{2\cdot 4\cdot 6}+\frac{x^8}{2\cdot4\cdot6\cdot8}+\cdots$

Find the sum of $$S(x)=\frac{x^4}{2\cdot 4}+\frac{x^6}{2\cdot 4\cdot 6}+\frac{x^8}{2\cdot4\cdot6\cdot8}+\cdots$$ What I did so far: It's trivial that ...
3
votes
4answers
54 views

Determine for which values of $z$ the series $\sum_{n=1}^{\infty} 2^{n}n^{n}z^{n}$ converges

For which values of $z$ does $\sum_{n=1}^{\infty} 2^{n}n^{n}z^{n} $ converge? I know the first step is to perform a ratio test to find the radius of convergence, but I'm having trouble choosing an ...
1
vote
1answer
27 views

series expressed in terms of other series?

Let us assume I have the coefficients $f_n$ and $g_n$ of: $$ f(x)=\sum_{n=0}^{\infty}f_n x^n\quad \text{and}\quad g(x)=\sum_{n=0}^{\infty}g_n x^n$$ Can I write?: $$\sum_{n=0}^{\infty}g_n ...
2
votes
1answer
40 views

What is the sum of the first 21 numbers of the formula below?

I know how to solve the geometric progression and arithmetic progression but this one seems strange to me, it isn't even a harmonic serie, Any help for solving it would be appreciated. ...
3
votes
1answer
50 views

Power Series Comprehension

Can anyone recommend any particularly clear, simple, yet thorough sources of explanation of Power Series in general? I'm a Calculus II student and am a complete beginner with infinite sequences and ...