1
vote
2answers
31 views

Finding the power series representation for $\ln(1 -10x)$ via integration.

I'm trying to find the power series representation for $ \ln(1-10x) $ Attempt at solution: $$ \ln(1-10x) = \int {-10\over1-10x} \ dx = -10 \int \sum_{n=0}^\infty (10x)^n dx $$ $$ = -10 ...
3
votes
2answers
54 views

Convergence radius of power series is infinite

Which function is given by a power series whose convergence radius is infinite? $$A. \ \ \ e^{-\frac{1}{x^2}}$$ $$B. \ \ \ \sin{\left(\frac{1}{x}\right)}$$ $$C. \ \ \ ...
0
votes
1answer
37 views

Taylor Series expansion and first four terms of $7x^2 e^{-4x}$

As the series I got $$ \sum_{n=0}^\infty (-1)^n(4x)^n/n! $$ which I think is right. However, I am not sure how to get the first four non zero terms.
0
votes
1answer
51 views

Radius and interval of convergence of the power series $\sum 2^{n^2}x^{n!}$?

How to calculate the radius and interval of convergence of the following series: $$\sum 2^{n^2}x^{n!}$$ The formula for the radius is: $$R = \frac{1}{\limsup_{n\to\infty} \sqrt[n]{|a_n|}}$$ or ...
3
votes
4answers
94 views

Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $

Evaluate: $$\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $$ I rewrote the sum as $$\sum_{n=0}^{\infty} \frac {1}{4^{8n-7}(8n-7)} - \sum_{n=0}^{\infty} \frac {1}{4^{8n-3}(8n-3)}$$ Now, I ...
-1
votes
2answers
23 views

Find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$

Let $\alpha$ and $\beta$ be two given constants, how to find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$, where $c_0 = {\alpha}^{\beta}$.
10
votes
1answer
201 views

Prove that $\exp(\log(\frac{1}{1-x})) = \frac{1}{1-x}$

I am trying to prove this directly by comparing the coefficients in the two series rather than using formal calculus. Here is what I have so far, but I think I made a mistake: \begin{align*} ...
2
votes
0answers
101 views

Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$

This question involves the following homework problem: PROBLEM Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form ...
0
votes
1answer
58 views

Question about infinitely many times differentiable function.

Could you please give me some hint how to solve this problem: Suppose $f(x)$ is infinitely many times differentiable function on R, $f(0)=f'(0)=f''(0)=0$. Prove : for all $A>0$ exists some ...
0
votes
1answer
35 views

Given a power series

Let c be a fixed number and consider the power series $\displaystyle\sum_{n=1}^ \infty \frac{c^{n-1}}{n} x^{n}$. a) Determine the convergence radius r for every value of $c \in \mathbb{C}$. In this ...
1
vote
1answer
26 views

Determine the maximal compact interval such that $\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1} = \arctan(x)$ holds true

The Assignment: Determine the maximal compact interval, such that the following identity holds true:$$\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1} = \arctan(x)$$ Explain your answer and show ...
3
votes
0answers
54 views

Prove that $e^{\ln{z}}=z$ from the power series

For my course in complex analysis we have to prove that the trivial relation $e^{\ln{z}}=z$. We are given the series for $\ln z$: $$f(w)=\sum_{n=0}^\infty (-1)^{n+1}\frac{w^n}{n}$$ $$\ln z = ...
2
votes
1answer
15 views

Radius of convergence of series with alternating coefficients

I need to compute, with proof, the radius of convergence $R$ for the series $$\sum_{k=0}^\infty \left(2-(-1)^n\right)^n z^n,$$ which is similar to a geometric series, except that the terms alternate ...
1
vote
2answers
55 views

A question about convergence interval of power series

Could you give me some hint how to solve this problem: Suppose $a_n$ is sequence defined as $a_1=\frac12,a_{n+1}=\frac12\left({a_n}^2+a_n\right)$. I managed to prove that $a_n$ is decreasing ...
2
votes
1answer
24 views

Power series solutions of differential equations, choosing x^n or x^(n+r)?

I cannot understand which one to use when solving differential equations by using power series solutions. For example in this question: Consider the following differential equation for $\alpha \in ...
1
vote
0answers
76 views

expand a rational function in a power series

$$\frac{4-x}{(2-x)(1-x)^2}$$ Expand in ascending powers of x, stating when the expansion is valid; also write down the coefficient of $x^n $
0
votes
1answer
21 views

Determining the first few coefficients of the complex power series of $\frac{z + 1}{(z + 2) \cos z}

As the title states, I'm trying to find the coefficients $a_0$, $a_1$ and $a_2$ of the power series $\sum_{n = 0}^\infty a_n z^n$ around 0 of \begin{align*} \frac{z + 1}{(z + 2) \cos z}, \end{align*} ...
2
votes
1answer
28 views

Radius of Convergence of Sum of two Series.

Hi all, I know there are similar questions on here, but none deal with the fact of trying to prove that $T \geq min\{R,S\}$. Intuitively this doesn't make sense to me, If you have, ...
2
votes
2answers
75 views

how to multiply infinite power series

I am doing an assignment for my precalculus 2 class. I am expanding two infinite power series and multiplying them together to prove that $\exp(ax)\exp(by) = \exp(ax+by)$ I'm not sure what I am ...
1
vote
3answers
31 views

Don't know why this power series representation is wrong…

I've run into something confusing. The problem is that I have to find the power series representation of $g(x)$ using the given $f(x)$, specifically $g(x) = \ln(1 - 3x)$ using $f(x) = \frac{1}{1-x}$. ...
4
votes
3answers
158 views

$\sin^2(x)+\cos^2(x) = 1$ using power series

In an example I had to prove that $\sin^2(x)+\cos^2(x)=1$ which is fairly easy using the unit circle. My teacher then asked me to show the same thing using the following power ...
0
votes
0answers
12 views

showing solution to kummer differential equation

struggling to solve kummer's differential equation and show that the confluent hyper geometric series is a solution. I have simplified the problem to showing that the sum over j to infinity of ...
1
vote
1answer
85 views

Solving differential equations using power series

I need to solve this differential equation by power series: $$y''+3xy'+(2x^{2}+6)y=0$$ Any help is great! Thanks!
2
votes
2answers
48 views

Difficulties evaluating the endpoints of the radius of convergence for a particular power series.

I am having difficulties evaluating the endpoints of the radius of convergence for the following power series. $$\sum_{k=0}^{\infty}\frac{(k!)^2 x^k}{(2k)!}$$ Using the ration test we get |x|<4. ...
0
votes
3answers
49 views

Problem about ODE and power series

For each $a \in \mathbb{Z}^+$ let the following ODE $$ x'' - \dfrac{a (a+1)}{(1 +t^2)} x = 0$$ Using power series around the origin, show that the equation has a solution $p_a(t)$ which is a ...
1
vote
1answer
54 views

Solve ODE using analytic solutions

Let the following ODE: $x'' + tx' + x = 0.$ Find the general solution $x(t) = a_0 x_1(t) + a_1 x_2(t),$ with $a_0, a_1 \in \mathbb{R}$ and $x_1(t), x_2(t)$ are $t$ power series convergent for ...
1
vote
3answers
277 views

First $3$ non-zero terms of the Maclaurin Series $\frac{1}{\sqrt{4+x^3}}$

Since each derivative will be multiplied by $3x^2$, are all the terms of this Maclaurin series $0$?
0
votes
1answer
62 views

Determine the sequence of coefficients $(a_n)_{n\in\mathbb{N_0}}$ so that: $\sum_{n=0}^\infty a_nx^n = \frac{e^x}{1-x} $

Assignment: Determine the sequence of coefficients $(a_n)_{n\in\mathbb{N_0}}$ so that $$\sum_{n=0}^\infty a_nx^n = \frac{e^x}{1-x}\ , \forall x\in\mathbb{R}: |x| < 1. $$ What I've got so far ...
1
vote
1answer
35 views

Interval of convergence homework

$$ \sum_{n=1}^{\infty} \left(\frac{-1}{4}\right)^n \frac{(5n)^n}{n!} (x-1)^n $$ First, I start with the ratio test: $$\lim_{n\to \infty} \left| \left(\frac{-1}{4}\right)^{n+1} ...
1
vote
1answer
102 views

Is there a power series which converges to $f(x) =| x|$ for all $x$?

I'm confused how to solve the following problem: "Is there a power series which converges to $f(x)$ = $\left| x\right|$ for all $x$?" Your help is greatly appreciated. Thanks a lot!
7
votes
3answers
214 views

2013th derivative of rational function

I am struggling to find $f^{(2013)}(0)$ for $$f(x) = \frac{1}{1 + x + x^3 + x^4}$$ I know that I should use power series, and following a hint I rewrote the equation as the following: $$1 = (1 + x + ...
1
vote
2answers
60 views

How do i convert $\frac{1}{2+x}$ to a summation?

I am given the summation for $\frac{1}{1-x}$. I get that I need to sub in $-x$ for $x$. I don't get how I am supposed to know where I put the $2$. I am not sure if there is a systematic procedure ...
-1
votes
1answer
133 views

Power Series Question.

By assuming a power series solution of the form $$y(x) = \sum_{m=0}^{\infty} c_mx^m , c_0 \not =0 $$ Show that; $$ 2y'+xy=x $$ has general solution $y(x)=1+Ae^{-x^2/4}$ where A is a ...
1
vote
1answer
86 views

Result of $\sum_{n=1}^{\infty} \frac{\sin(nx)}{n}$

I already showed that $\displaystyle g(x)=\sum_{n=1}^{\infty} \frac{\sin(nx)}{n}$ converges uniformly on $[\delta,2\pi-\delta ]$ for $\delta>0$ Now I have to calculate the limit of ...
1
vote
1answer
160 views

Introduction to Analysis: Multiplication Theorem for Series

I've been stuck on this problem over the weekend so I decided to ask for some direction. The problem reads: "The multiplication theorem for series requires that the two series be absolutely ...
0
votes
0answers
132 views

Proving Convergence of a Power Series With Partial Sums

Let $\sum_{0}^{\infty} a_n$ be a series and $s_n$ its sequence of partial sums. Suppose $\sum a_nx^x$ converges for $|x| <1$; let $f(x)$ be its sum. Then $\displaystyle \sum_{0}^{\infty}s_nx^n = ...
2
votes
2answers
376 views

Radius of Convergence of $\sum ( \sin n) x^n$.

Thank you very much in advance for any assistance/advice on solving this problem. I am fairly new to power series and determining the radius of convergence. Determine, with proof, the radius of ...
0
votes
1answer
42 views

Problem with power series problem.

Using this theorem: Let $(a_k)$ be a sequence in $\Bbb R$ and let $x_0\in \Bbb R$: The power series: $$\sum_{k=0}^\infty a_k(x-x_0)^k \qquad \text{and} \qquad \sum_{k=0}^\infty ...
0
votes
1answer
31 views

As resolved the identity of this series?

Use the identity $\cos((k-\frac{1}{2})x) - \cos((k+\frac{1}{2})x) = 2\sin kx \sin \frac{x}{2}$ to show that $S_n:=\sum_{k=1}^{n}\sin kx=\frac{1}{2\sin ...
1
vote
2answers
62 views

What are the values ​​for which the series converges?

Determining the values ​​of $a$ and $b$ so that the series $\sum a_n$ converges, where $$a_n=\ln n-a\ln(n+1)+b\ln(n+2)$$
0
votes
1answer
53 views

Simple differentiation question that I am unsure about

I am in the process of re-learning differentiation and am stuck on this as part of a larger problem. Can you explain to me why when differentiated 4 times this: $$y = \sum_{n=0}^{+\infty} ...
1
vote
1answer
121 views

Finding coefficients of a differential equation represented by power series

I am studying for a discrete mathematics exam and have gotten stuck on this question: Any function y of a real variable x that solves the diff erential equation: $$\frac{d^4y}{dx^4} -16y =0$$ may ...
1
vote
2answers
42 views

General case of radius of convergence of a power series

Show that if the series $\sum_{n=1}^{\infty} a_nx^n$ has a radius of convergence $L = R$ so the series $\sum_{n=1}^{\infty} a_nx^{kn}$ has radius of convergence $L = R^{\frac{1}{k}}$. Anyone could ...
1
vote
2answers
122 views

Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$ [duplicate]

Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$. I have no idea to solve this problem. Anyone could help me?
2
votes
1answer
80 views

Interval of convergence $\sum_1^\infty \frac{2^n}{3n}(x+3)^n$

$$\sum_1^\infty \frac{2^n}{3n}(x+3)^n$$ I do the ratio test to find the radius. $$\frac{2^{n+1}}{3(n+1)}(x+3)^{n+1} *\frac{3n}{2^n (x+3)^n}$$ This should reduce down to $2|x+6|< 1$ This is ...
0
votes
1answer
64 views

Interval of convergance $\sum_0^\infty \frac{(2n)!}{(n!)^3}*x^n$

$$\sum_0^\infty \frac{(2n)!}{(n!)^3}*x^n$$ I have no idea how to do this. I tried to write it all out with the ratio test and I get some weird expression that doesn't make sense like $$x * ...
2
votes
2answers
74 views

Let $f(x) = \int \frac{x}{1-x^{8}}dx\,$

Let $f(x) = \int \frac{x}{1-x^{8}}dx\,$. Represent $I(x)$ by a power series $\sum^{\infty}a_{n}x^{n}$.(Find $a_{n}$) What is the radius of convergence of $I(x)$ ? Two curves are generated by polar ...
1
vote
1answer
43 views

Rearranging power series expansion to get parameter on denominator

How can we rearrange $$T=\dfrac{k V+g}{gk}\bigg(kT-\dfrac{1}{2}k^{2}T^{2}+\dfrac{1}{6}k^{3}T^{3}\bigg),$$ to get $$T=\dfrac{2V/g}{1+k V/g}+\dfrac{1}{3}k T^{2}$$ ?
1
vote
1answer
230 views

Radius of convergence of power series (complex)

I don't know if my reasoning is right on this exercise: If the power series $\sum a_n z^n$ has radius of convergence $R$, which is the radius of convergence of the series $\sum a_n^2 z^n$ and $\sum ...
2
votes
1answer
56 views

What happens outside radius of convergence

A real power series $\sum_{n=0}^\infty a_n z^n$ has radius of convergence $R$. I am able to prove that for any real number $r>R$, the sequence $|a_n|r^n$ must be unbounded. Must it also tend to ...