2
votes
2answers
60 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
29 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 $f(x) = \frac{1}{1-x}$. ...
3
votes
3answers
115 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
67 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
43 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
44 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
51 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
157 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
32 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
101 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
209 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
58 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
84 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
149 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
126 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
303 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
61 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
52 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
114 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
34 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
117 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?
1
vote
1answer
69 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
59 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
73 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
41 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
198 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
52 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 ...
1
vote
3answers
49 views

Which one is the correct series expansion?

Is $$p^{n+1} = p^0+p^1+ \dots + p^n$$ or $$p^{n+1} = p^0\times p^1\times \dots \times p^n\text{ ?}$$ I am confused. please explain the correct one.
1
vote
1answer
53 views

Homework: Maclaurin Power Series Help

I'm trying to find the Maclaurin Power Series for $$f(x)=\frac{3x-8}{3x^2+5x-2}$$ but each degree of differentiation gets more complex with no discernible pattern. Any help is appreciated, thanks.
3
votes
1answer
47 views

Struggling to understand a couple of concepts with series

I have two questions: neither of which are homework problems but certainly pertain to my ability to do the homework. The first regards the harmonic series. The question has been answered often here ...
1
vote
1answer
50 views

Functions $f, g$ are given. We know that we can expand them into power series around $x_0=0$, they also satisfy: $f(\frac{1}{k})=g(\frac{1}{k})$ …

Functions $f, g$ are given. We know that we can expand them into power series around $x_0=0$, they also satisfy: $f(\frac{1}{k})=g(\frac{1}{k})$ for sufficiently large $k \in \mathbb{N} $. Prove that ...
2
votes
2answers
2k views

Finding the Laplace Transform of sin(t)/t

I'm in a Differential Equations class, and I'm having trouble solving a Laplace Transformation problem. This is the problem: Consider the function $$f(t) = \{\begin{align}&\frac{\sin(t)}{t} ...
0
votes
2answers
60 views

Show that cosh(2) is between two values.

I'm reviewing for exams and this question has got me stumped: Show that: $3\dfrac{2}{3} \leq \cosh(2) \leq 3\dfrac{2}{3} + 0.1$ I've determined the series form of cosh(x) to be: ...
0
votes
1answer
69 views

Demonstrating the coefficients of the power series of $\frac{1}{1-z-z^2}$ satisfies a recurrence relation.

I have the power series $$\frac{1}{1-z-z^2} = \sum_{n=0}^{\infty} c_nz^n$$ and I'd like to show that the coefficients of this power series satisfy $c_n=c_{n-1}+c_{n-2}$. I thought the most obvious way ...
1
vote
5answers
956 views

Use Cauchy product to find a power series represenitation of $1 \over {(1-x)^3}$

Use Cauchy product to find a power series represenitation of $$1 \over {(1-x)^3}$$ which is valid in the interval $(-1,1)$. Is it right to use the product of $1 \over {1-x}$ and $1 \over ...
1
vote
1answer
41 views

Trying to show that the product of two power series equals 1.

I've reduced a large homework problem to the following smaller problem. Let $P = \sum_{i=0}^\infty a_i X^i$ denote a formal power series over a field. Assume $a_0 \neq 0$, and define $Q = ...
2
votes
1answer
2k views

Maclaurin Series for $\arctan(x)$ by successive differentiation

I am trying to find a Maclaurin Series for $\arctan(x)$ up to the term with the fifth power of x and I have to use the method of successive differentiation. I know (from an example in my notes) the ...
1
vote
0answers
95 views

Relating terms in differential equation with power series

Having problems with a task on a differential equation containing a power series. Given $$\frac{dx}{dt} = \lambda x + \sum_{n=2}^\infty b_n x^n$$ $$\frac{dy}{dt} = \lambda y$$ $$x(y) = y + ...
0
votes
1answer
113 views

Complex analysis Laurent series evaluated on unit circle

Let $f(z)$ be a function analytic on an annulus that includes the unit circle $z=e^{i\theta}$. By taking that circle as the path of integration for the coefficients in the Laurent series, show that $$ ...
2
votes
0answers
94 views

Properties of Entire Functions

a). Suppose an entire function f is bounded by M along $\vert z \vert = R$. Show that the coefficients $C_k$ in its power series expansion about $0$ satisfy $ \vert C_k \vert \leq \frac{M}{R^k} $. I ...
5
votes
3answers
104 views

Representing Functions as Power Series

Rewrite $$f(x)=(1+x)/(1-x)^2$$ as a power series. Work thus far: I separated it into two parts: $$1/(1-x)^2 + x/(1-x)^2$$ I realize that the first expression is the derivative of $1/(1-x)$ and ...
-1
votes
5answers
138 views

Expanding the power series

$$g_2(\epsilon^1 \phi_1+ \epsilon^2 \phi_2+ \epsilon^3 \phi_3+\cdots)^2+ g_3(\epsilon^1 \phi_1+ \epsilon^2 \phi_2+ \epsilon^3 \phi_3+\cdots)^3+g_4(\epsilon^1 \phi_1+ \epsilon^2 \phi_2+ \epsilon^3 ...
1
vote
1answer
86 views

What is the radius of convergence?

$\displaystyle\sum_{n=1}^{\infty}x^{2n-1}/a_{n}$ What is the radius of convergence? Ps: I found that $\limsup|1/a_{n}|^{1/n}=1/6$ But I am confused because of $x^{2n-1}$ What is the radius of ...
1
vote
1answer
18 views

What values does the function $Z(y)$ have at various interval?

When $y\leq0$; $H(y)=0$ When $y>0$; $H(y)=e^{-\dfrac1y}$ What values does the function $Z(y)$ have at various interval? Where $Z(y)=H(1-y)(1+y)$ Please show this!
2
votes
1answer
62 views

Taylor series representation of a function.

I'm working on expressing the function $f(x)=\frac{6}{x}$ as a taylor series about $-4$. I've got the general idea, but I'm not quite there yet. I've come up with the equation ...