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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2
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3answers
27 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
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0answers
27 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
23 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
18 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
13 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
32 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 ...
1
vote
0answers
41 views

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 ...
2
votes
1answer
32 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
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3answers
25 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
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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
29 views

Test the uniform convergence of the series in indicated region

Test the uniform convergence of the series $$\sum_{n=1}^ \infty (-1)^{n-1} \frac{z^{2n-1}}{1-(z^{2n-1})}\quad |z|\lt 1$$ How can I do?
0
votes
1answer
33 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
21 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
39 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
44 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
32 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
63 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
19 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 $n \ge 1$.
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0answers
14 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
1answer
185 views
+50

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
54 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
30 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. ...
-1
votes
0answers
30 views

Use Abel's Theorem to test the uniform convergence

Use Abel's Theorem : Test the uniform convergence of the series $$\sum_{n=1}^ \infty\frac{1}{z^2-n^2\pi^2}$$ $\forall z\not= \pm \pi, n\in\Bbb N $
0
votes
0answers
18 views

Test the uniform convergence :-

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 uniform convergence of the series ...
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
21 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
46 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
39 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
133 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: ...
1
vote
2answers
33 views

Question about Power Series

I have this problem: $$\sum_{n=0}^{\infty}\ 64^n\ (x-10)^{3n+1}$$ After using the Ratio Test I am left with this: $$ \lim_{n\to\infty}\big|\frac{64^{n+1}(x-10)^{3n+4}}{64^n(x-10)^{3n+1}}\big| $$ ...
0
votes
0answers
25 views

Let $f(z) = \sum_{k=0}^\infty\left(\frac{k^3}{3^k}\right)z^k$, compute each of the following.

Let $f(z) = \sum_{k=0}^\infty\left(\dfrac{k^3}{3^k}\right)z^k$, compute each of the following. $$f^{(6)}(0)\tag{a}$$ $$\text{And }\oint_{|z|=1}\dfrac{f(z)}{z^4}dz\tag{b}$$ I have no idea how to ...
0
votes
2answers
40 views

Problem understanding notation

I'm learning about generating functions and in the opening explanations my book (and various sources) claim: $$a_n = 1 \forall n \in \mathbb{N}_0, \ \ \ f(x) = \frac{1}{1-x}$$. I read this as: ...
0
votes
0answers
30 views

prove that the series convergent by ($\epsilon$-$N$)

I want to prove that the following series for $e^z$ is absolutely and uniformly convergent for all values of $z$ : $1+z+\frac{z^2}{2!}+ ... +\frac{z^n}{n!}+...$ Thanks For All
1
vote
2answers
34 views

Interval of Converge for a Power Series

I would really appreciate some help on this problem that I have been working on. It's a power series question. I have to find the interval of convergence for this power series. The power series is: ...
0
votes
2answers
48 views

proving series convergence by definition

I have a question in the homework, and I tried to prove it but I don't know if it is correct? I want to prove that the series $$ \sum_{n=1}^{\infty} \frac{z^n}{n(n+1)} $$ is absolutely convergent for ...
1
vote
2answers
56 views

Find the power series for $f(x) = \frac{\cos(x^3)}{2x^2}$

I'm pretty sure if it were just $\cos(x^3)$ i could subsititue $x^3$ for $x$, everywhere in the known series, but what do I do because it's divided by $2x^2$?