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).

learn more… | top users | synonyms

0
votes
0answers
21 views

Find the Intervel of Convergence of the Power Series

One question for you guys. I need to find the interval of convergence of the power series. Having a lot of problems with this one and would love a thorough explanation, however any help is ...
3
votes
3answers
113 views

Help with radius of convergence of a power series.

I need to determine the radius of convergence of the series $\sum_{n=1}^\infty a_nx^n$, where $a_n=a^n+b^n$ and $a,b$ are real numbers. Not sure how to approach this one.
1
vote
0answers
14 views

Combining even and odd parts of a Chebyshev series

I imagine this will be an easy problem, perhaps even routine, for some. I am learning to manipulate sums and need insight. I started with a power series $$s(x) = \sum_{n=0}^{\infty} a_n x^n$$ and ...
4
votes
1answer
52 views

Prove that $\frac{1}{\sqrt{1-z}}=\sum_{n=0}^{\infty}\frac{1}{4^{n}}\binom{2n}{n}z^{n}$ using Cauchy product

need to prove using Cauchy product for series for all $\left|z\right|<1$ that $$\frac{1}{\sqrt{1-z}}=\sum_{n=0}^{\infty}\frac{1}{4^{n}}\binom{2n}{n}z^{n}$$ (with appropriate branch of the root ...
1
vote
0answers
25 views

Find the power series for a definite integral

I am a bit unsure when integration is used together with summation. Here is my question: Find power series for $\int_0^{1} \frac{\sin x}{x}dx$ in the form $\sum_{k=1}^{\infty} a_kx^k$ Here is what I ...
0
votes
0answers
14 views

Taylor expansion and expansion in powers of z-1

I am trying to expand $z^2/(z+1)^2$ as a Taylor Series. I have acquired its partial fraction decomposition of $z^2/(z+1)^2$ = $(1/6)*(1/(z+1)) + (5/6)(1/(z-5))$. The first term is in the form ...
1
vote
1answer
60 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
2
votes
2answers
70 views

Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$

Does anyone know the sums of the following two series? $$\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$$ $$\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{4n-1}}{4n-1}$$ I encounter such series in my work.
0
votes
2answers
40 views

Uniform convergence of the series

Test the uniform convergence of the series $$ \sum_{n=1}^\infty \frac{1}{z^2 - n^2 \pi^2}$$ $$ \forall z \not= \pm n\pi,\;\; where n \in\mathbb N$$ Can I find $M_n$ such that $$ ...
2
votes
3answers
39 views

Finding power series of function

could anyone help me answer question? $$F(x)=\ln\left(\dfrac{7+x}{7-x}\right)$$ Find a power series representation for the function.
2
votes
0answers
31 views

Fractional Derivatives

If we define the (forward) difference operator as $$\Delta f(x)=f(x+\Delta x)-f(x)$$ we can break it up using the "shift" operator $E\,f(x)=f(x+\Delta x)$ and the "identity" $1\,f(x)=f(x)$. Then ...
0
votes
2answers
28 views

How to expand 1/(1+z^2) in powers of (z-a)?Here z is a complex number.

How to expand 1/(1+z^2) in powers of (z-a)?Here z is a complex number. I know for people who knows how to do this this is a stupid problem.But I am just a beginner.Differentiating 1/(1+x^2) seems not ...
0
votes
2answers
20 views

Find the order of a function.

Consider the function $(x + 2)\cos^2 x$. Determine its order in terms of big-O notation. (A) $O(x)$ (B) $O(x^2)$ (C) $O(\log (x))$ (D) None of the above
1
vote
1answer
21 views

Power series convergence of random walk transition matrix

I would like to find out if $$ \sum_{t=0}^\infty P^t = \left( I- P \right)^{-1} ~,$$ where $P = D^{-1}W ~ $ is a random walk transition matrix. $W$ is a matrix describing weights in a graph and ...
0
votes
1answer
33 views

Power series approximation

Hi does anyone knows how to solve this question. Use power series to approximate the definite integral to within the given accuracy $\int_{0}^{1}x^{2}\sin(x^{4})dx$ Error $<0.001$ I managed to ...
5
votes
1answer
144 views
+200

Take 2: When/Why are these equal?

This didn't go right the first time, so I'm going to drastically rephrase the query. As per this previous question, I am wondering if the two series ...
3
votes
1answer
35 views

When are these series equal?

Suppose we have a power series $$\sum_{n=0}^\infty {a_nb_nx^n}$$ When is it true that the series obtained by eliminating $b_n$ is proportional to the original series? $$\sum_{n=0}^\infty ...
0
votes
3answers
27 views

Finding a special power series

Find a power series for F, such that $F'(x)=e^{-x^2}$. Don't understand how to come up with the solution
1
vote
1answer
13 views

Explanation on how to turn a numerical sequence into a power serie

I'm taking a calculus class, but I skipped school the past week due to health problems. I spoke to my teacher and classmates and they told me that they had seen power series topic. So I got a copy of ...
1
vote
0answers
33 views

Test the uniform convergence of the series in indicated region

Test the uniform convergence of the series I tried to find $M_n$ such that $|\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}|\le M_n $ by using Abel's Theorem This is the question : Test the ...
0
votes
1answer
34 views

Taylor series convergence with natural logs

I am working on this problem. Find Taylor series of function $f(x)=\ln(x)$ at $a = 6$. $$f(x) =\sum_{n=0}^\infty c_n (x- 6)^n$$ I seem to be having trouble with the interval of convergence can ...
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 ...
0
votes
1answer
22 views

Differential equation by series solution method: equating coefficients to zero

I am following the solution for a problem, and I am stuck at the following equation: $$2a_2+\sum_{n=1}^\infty \left[(n+2)(n+1)a_{n+2}-a_{n-1}\right]x^n=0\tag1$$ Now, the professor equates the ...
1
vote
1answer
44 views

How to justify, $\sum_{n=1}^{\infty} a_{n} x^{n} - \sum_{n=1}^{\infty}a_{n}y^{n}=\sum_{n=1}^{\infty} a_{n} (x^{n}-y^{n})$?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$ and we let $K_{1}$ be a compact subset of ...
0
votes
1answer
24 views

What is the significance of finding the series solution of a differential equation “about a point”?

I am learning the series solution method of solving differential equations, and I am curious as to what the rationale is for finding out the solution of the equation about a particular point. It seems ...
0
votes
2answers
41 views

Power series with $f(x)=\frac {1}{1+100x^2}$

I am working on the power series. Here is the question $$f(x)=\frac {9}{1+100x^2}$$ represented as a power series $$f(x) = \sum^{\infty}_{n=0}c_nx^n$$ I need to find $c_0,c_1,c_2,c_3,c_4,R$ I got ...
2
votes
2answers
45 views

Is it true that, $|e^{x}-e^{y}|\leq C \cdot |x-y|$?

Define $f:\mathbb R \to \mathbb R$ such that $f(x)= e^{x}-1:= \sum_{n=1}^{\infty} \frac{x^{n}}{n!};$ for $x\in \mathbb R.$ My Question: Can we expect $|f(x)-f(y)|\leq |x-y| \cdot C;$ where $C$ is ...
2
votes
2answers
33 views

Finding Function of Series: $e^{-kx}$

If the series representation of $e^{-x}$ is: $$\sum_{k=0}^{\infty} \frac{(-x)^k}{k!} $$ Then what is for $e^{-kx}$?
3
votes
0answers
66 views

Question about Big O notation for asymptotic behavior in convergent power series

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
2
votes
0answers
34 views

Question about Big O notation for asymptotic behavior in convergent power series [duplicate]

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
0
votes
1answer
20 views

Coefficient of power series when $p(x) = \sum b_nx^n$ converges for $|x| \le 1$ and $p(x) = 0$ for $|x| \lt \delta$.

Suppose that the power series $p(x) = \sum b_nx^n$ converges for $|x| \le 1$. Suppose that for some $\delta \gt 0 , p(x) = 0$ for $|x| \lt \delta$. Show that $b_n = 0$ for all $n \ge 1$.
0
votes
0answers
16 views

conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...
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 $f(x) = \frac{1}{1-x}$. ...
2
votes
3answers
251 views

if $ S(x)=\sum_{n=0}^{\infty}a_{n}x^n,|x|<R$, $S_k(R)$ bounded,prove or disprove $\lim_{x\to R}S(x)$ exist?

let $$S(x)=\sum_{n=0}^{\infty}a_{n}x^n,a_{n}>0,|x|<R$$ (or mean that powr series have radius of convergence R.) and let ...
0
votes
1answer
18 views

A integral about Powers of x and binomials.

Here is the integral $$\int_0^\infty {\frac{{{x^{p - 1}}}}{{x + a}}{{\left( {bx + c} \right)}^q}} dx,where{\text{ }}a,b,c > 0,p,q \geqslant \frac{1}{2}$$
0
votes
1answer
14 views

Radius of convergence of $\sum a_nx^n$ where $a_n = {k \choose n}$

Consider the power series $\sum a_n x^n$ where $$ a_n = {k \choose n} $$ for some $k$. What is the radius of convergence of this power series? I got one. Does that seem correct? I got that the ...
6
votes
2answers
56 views

What is the Taylor series of $\frac{1}{\sin(z)}$ about $z_0 = 1$?

This was a exam question so I know it cannot take too long to write out the proof. Only I cannot see an answer. I would imagine you write $\sin(z) = \sin(1+(z-1)) = \sin(1)\cos(z-1) + ...
0
votes
5answers
33 views

Finding sum of Power series

Hi could anyone help me with this question Determine the sum of the power series: $$S=-\sum_{n=1}^{\infty}\frac{(1-x)^n}{n}$$ Where x=1.74 I tried to differentiate this expression, but I do not ...
1
vote
1answer
28 views

Determine the value of r where the series converges

show that $$ \big(r\big)^{ln(n)} = \big(n\big)^{ln(r)} $$ Then determine the values of r (with r>0) for which the series $$ \sum_1^\infty (\big(r\big)^{ln(n)})$$ converges. r must be in what ...
0
votes
1answer
22 views

Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
0
votes
3answers
55 views

What is the sum notation equivalent to this series?

I tried to solve this question Find radius of convergence of the series $$1+ \frac{a.b}{1.c}z + \frac{a(a+1)b(b+1)}{1.2...c(c+1)}z^2+.... $$ The book answer is 1 I cant obtain the nth term in the ...
0
votes
2answers
31 views

find region of convergence of the series

I tried to solve this question Find domains of convergence of the series $$\sum_{n=1}^ \infty \frac{z^n}{n (\log n)^2 }$$ How can I do this .
1
vote
2answers
22 views

Find domains of convergence of the series

I tried to solve this question but the final solution which I obtain is not the as same as in the text book Find domains of convergence of the series $$\sum_{n=1}^ \infty ...
1
vote
2answers
19 views

Calculate ch(0.2) to the nearest 0.01

Help me calculate ch(0.2) to the nearest 0.01. I tried to rewrite ch as a series but I still don't know how to evaluate it and what to do with factorial Help me please. it's very important
0
votes
3answers
48 views

Find a radius of convergence of power series

I have to Find a radius of convergence of this power series I' ve decided to use D'alambert indication: Looking for a limit i meet a problem with a factorial Please. help me finish this ...
0
votes
1answer
22 views

Power Series to solve non linear differential equations.

I've been revising Power series recently and their application when it comes to solving linear differential equations, but in this question I'm not sure what to do when it's a non linear function. I ...
0
votes
1answer
39 views

Express the coefficients of $\exp(f(x))$ using the coefficients of $f(x)$

Given a formal series $$f(x)=\sum_{k=1}^\infty f_k x^k$$ what is $$K_n:=\left[\left(\frac{d}{dx}\right)^n e^{f(x)}\right]_{x=0}$$ in terms of the coefficients $\{f_k\}$? I stumbled upon this ...
2
votes
0answers
40 views

Replacing $q^2$ by $q$

I have a rather strange question. Suppose we are given a formal power series $$S(q^2) = \sum_{n = 0}^\infty a_n q^{2n}.$$ I wish to replace $q^2$ by $q$. This implies that $S(q) = \sum_{n = 0}^\infty ...
1
vote
2answers
151 views

What is $\sum_{n=0}^{\infty}|a_nz^n|^2=\frac{1}{2 \pi}\int_{-\pi}^{\pi}|f(ze^{it})|^2dt$ for?

Let $f : \mathbb C\rightarrow \mathbb C$ be an analytic function : $f(z)= \sum a_n z^n$ It holds that $$a_n z^n= \frac{1}{2 \pi}\int_{-\pi}^{\pi}f(ze^{it})e^{-int}dt$$ and ...
1
vote
2answers
19 views

Find a power series by comparing it to a geometric series?

Find the power series for the following function: $f(x)=\frac{1-x}{x-3}$ centered at x=1. This is what I've done: ...