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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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
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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
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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
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2answers
34 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
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0answers
26 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
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2answers
41 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
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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
2
votes
2answers
38 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
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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
59 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 $2x^2$?
1
vote
2answers
67 views

Is there any closed form for this series?

It's a power series that I found during the computation for my research. \begin{equation*} \sum_{k=0}^n \binom{n}{k}\frac{n!}{(n-k)!}x^{n-k}(-1)^k. \end{equation*} Without the annoying term of ...
1
vote
1answer
17 views

Nth Term Proof Taylor Polynomial

suppose that Msub_n(x) is the nth order maclaurin polynomial for f(x). Show that if k is a constant then Msub_n(kx) is the nth-order maclaurin polynomial for f(kx)
0
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2answers
28 views

Let $f(z) = \sum_{j=0}^{\infty}a_jz^j$ be the Maclaurin expansion of a fnction $f(z)$ analytic at the origin. Prove each of the following statements.

Let $f(z) = \sum_{j=0}^{\infty}a_jz^j$ be the Maclaurin expansion of a function $f(z)$ analytic at the origin. Prove each of the following statements. $(a)$ $\sum_{j=0}^{\infty}a_jz^{2j}$ is the ...
1
vote
1answer
23 views

Find the power series for $d/dx(\arcsin x)$

How would you find the general power series for $\frac{1}{\sqrt{1-x^2}}$ , without using the general rule for arcsinx? I understand it is necessary to use binomial series, but I am having trouble ...
1
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0answers
24 views

Solution regarding Power Series and ODE's

About 4 months ago I posted Series solution to $y''-xy'-y=0$. I ran through the analysis and it appeared that I solved the ODE . The solution seemed to be ...
2
votes
2answers
14 views

Radius of convergence two power series (by using Cauchy test).

Let power series $\sum_{n=0}^{\infty} a_nz^n$ have radius of convergence $R$. I would ask you, is it true that $\sqrt[n]{a_n} \rightarrow \frac{1}{R}$? If it is true, then power series ...
1
vote
1answer
29 views

Power series of $f(z) = \frac{z}{1-z}$

Find power series of $f(z) = \frac{z}{1-z}$ in point $z_0 = i$ and find radius of convergence this power series. Of course, I can find $f^{(n)}(z_0)$ and then I will have $$f(z) = \sum_{n=0}^{\infty} ...
0
votes
3answers
38 views

Why $\sum_{k=1}^n (\frac56)^{k-1}\cdot (\frac16)=1-(\frac56)^n$

Why $$\sum_{k=1}^n \left(\frac56\right)^{k-1}\cdot \left(\frac16\right)=1-\left(\frac56\right)^n$$?? Which formula I need to use to calculate it? Thank you!
0
votes
1answer
21 views

Maclaurin series and general terms

For a general term in a maclaurin (or any other) series e.g. x^r/r! is it always that r=0, or r=1 at the first term or does it have nothing to do with the term it appears in e.g. r=10 could come ...
0
votes
2answers
36 views

Represent Power Series of a function

Hi could anyone help me answer this question Find the power series representation for the function and determine the radius of convergence $f(x)=\frac{x^2}{\left(1-2x\right)^2}$ After getting ...
1
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2answers
20 views

Power series of a function about a non zero point

No clue how to ask questions here so here goes nothing! How do I work towards finding the power series of a function centered about a point a not equal to $0$? The specific question I was asked is to ...
0
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2answers
42 views

Finding the Function of a Power Series: $\sum kx^{k+1}/3^k$

Given: $$\sum_{k=1}^{\infty} \frac{kx^{k+1}}{3^k}$$ Im guessing its equivalent to: $$\sum_{k=1}^{\infty} k\left(\frac{-1}{3}\right)^k x^{k+1}$$ But I am not sure on how to advance past this step. ...
2
votes
1answer
46 views

Convergence of a series that looks similar to $e^x$

Suppose I have some $\epsilon > 0$ and some constant $c > 0$. Does the series $$ \sum_{n=1}^{\infty} \frac{c^{n^{\epsilon}} }{[n^{\epsilon}]!}, $$ where $[r]$ is the integral part of a real ...
0
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0answers
43 views

How to determine undetermined coefficients of infinite series solution of ODE by seeking the solutions that vanish at infinity?

For an ODE like $f''+p(x)f'+q(x)f=r(x)$ where $p,q,r$ are analytic functions I'm trying an infinite series solution $f=\sum_{n=0}^\infty a_n x^n$. All coefficients $a_n$ can be obtained in terms of ...
3
votes
3answers
118 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
2answers
46 views

Retrieving $f(x)$ from Taylor Series

I've been trying to extract the $f(x)$ from the following Taylor series: $$ \sum_{k=1}^{\infty} (-1)^{k}\frac{kx^{k+1}}{3^{k}} $$ I moved things around a bit to make it look like this: $$ ...
0
votes
1answer
22 views

Find the radius of convergence and interval of covergence of the series.

I'm in the process of trying to learn about the topic of series, power series, etc. (when I say learn I mean self-teach... everything urgh). My general understanding of the topic of Series as a whole ...
1
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0answers
41 views

Calculus(power series)

Hi could anyone help me solve this problem.(without expanding the function). I cant seem to get it. Express the function as a power series $$\ln\left(\frac{1+x}{1-x}\right)$$ I managed to split the ...
2
votes
1answer
96 views

How to estimate the growth of a “savage” function near 1?

Say I have a function which exists within the unit disk, say $$f(x)=a_0+a_1x+a_2x^2+...$$ If we know sufficient information about the coefficients, say we know the growth rate of ...
0
votes
1answer
34 views

Proving that the derivative of the integral of a power series equals the original power series

I've been thinking about the following recently: If we have a power series $f(x) = \sum_{n=0}^\infty a_n(x-c)^n$ and $F(x)=\sum_{n=0}^\infty \frac{a_n}{n+1}(x-c)^{n+1}$ where $F(x)$ is constructed by ...
3
votes
2answers
50 views

Finding terms of a Taylor series where $f(x)$ is a function with a power

I've been stuck with this Taylor series problem for a while now. We have that $$ f(x) = (1 + x^2)^{-2/3} $$ and it's centered at $0$. So what I thought of doing was the $$ \frac{f^{n}(a)(x - ...
2
votes
1answer
37 views

Radius of Convergence of product of power series

Is the following statement true? If $P(z)$ is a power series over $\mathbb C$, then $ P(z) $and $P(z)^n$ have same radius of convergence for any positive integer n.
1
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3answers
30 views

A question about sum of probability… if $P(X\ge n)-P(X=n)=P(X>n)$

If we know that $P(X\ge n)=(1-p)^{n-2}$ (This is not the main subject of the question, so I wont explain about it, hope this OK, but in sort: we get it because $P(X\ge n)=\sum_{k=n}^\infty ...
0
votes
1answer
28 views

What is the sum of this series: $\sum_{k=n}^{\infty}(1-p)^{k-2}\cdot p$?

I have: $$\sum_{k=n}^{\infty}(1-p)^{k-2}\cdot p$$ $|p|<1$. The sum is: $$=\frac{(1-p)^{n-2}\cdot p}{1-(1-p)}=(1-p)^{n-2}$$ Or I wrong? I use the fact that the sum of a series is: $\frac{a}{1-p}$ ...
0
votes
1answer
59 views

Complex Power series with factorials

Find the radius of convergence of $$\displaystyle\sum_{n=0}^\infty z^{n!}$$ $$\displaystyle\sum_{n=1}^\infty {(-1)^nz^{n(n+1)}}/{n}$$ What is the behavior of the series for $z=1, -1,i$
1
vote
3answers
192 views

Find complicated Taylor Series

According to some software, the power series of the expression, $$\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}}$$ around $x=0$ is $$\sqrt{x}-x^{3/2}+\mathcal{O}(x^{5/2}).$$ When I try to do it I find that I ...
0
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0answers
30 views

Reference for power series

I would need some references for power series, Taylors series of elementary functions, derivation and integration of power series, convergence of sequences of functions and series of functions. The ...
5
votes
2answers
65 views

Closed form of generating function consisting of power of two binomials

Let $g(x)$ be infinite formal power series and $$g(x) = (1 + x)(1 + x^2)\cdots(1 + x^{2^k})\cdots$$ Show that $(1 - x) g(x) = 1$. My book gives following proof: Using a fact that $(1 - x^k)(1 + ...
0
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2answers
76 views

How do I find the coefficients in this product?

$(x+x^2+x^3+x^4+x^5+x^6)^5$ How do I find the coefficients of this product quickly?
0
votes
1answer
31 views

How to Expand $f(x) = \sqrt{x^2-y^2}$ about $x=y$

How would one go about expanding $f(x) = \sqrt{x^2-y^2}$ about $x=y$? You can't do a Taylor series, because $f'(x=y)$ diverges. I tried a few other things, but to no avail. Interestingly enough, ...
0
votes
2answers
32 views

Convergence of complex power series question

I need some help to solve this problem and find the domain of convergence of the following power series: $$\displaystyle\sum_{n=0}^\infty(2^n+i^n)(z-2i)^n$$ Thank you!
0
votes
1answer
23 views

Radius of convergence problem

I need a hint on this problem, been staring on the blackbord a long time now. Problem: Suppose $f(x) = \sum a_{n}(x-x_{0})^n$ has radius of convergence $R$ and $0 < r < R_{1} < R$. Show that ...
1
vote
2answers
41 views

polynomial series and root multiplicity

Excuse me, because I know this is a double post but I can't for the life of me find the original post. Given a sequence $(a_n)$, one can construct a polynomial of the form ...
3
votes
1answer
69 views

Sum of combinations series

What is the value or tight upper limit of the following summation: $$\sum_{k=0}^n{n\choose k} x^{k(n-k)}$$
0
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2answers
50 views

Limits of nth-root on various sequences

I have been computing the ratius of convergence for different power series using the the root-test. In many of them, I am left with nth-root expressions that computationally approaches 1 when n goes ...
1
vote
2answers
49 views

Power Series Representation…

I am having a hard time understanding how to proceed with this question... Find a power series representation for the function and determine it's radius of convergence $$ f(x)= x^2\ln(1+x^2) $$ How ...
1
vote
2answers
38 views

Simplifying ratio test with exponents $k+1$

Question: Find the interval and radius of convergence. $$\sum_{k=1}^\infty\frac{(x-1)^k(k^k)}{(k+1)^k} .$$ I applied the ratio test. ...
0
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0answers
23 views

derivatives of a composite function

There exists a 'closed-form' formula for the higher order derivatives of a composite function $f\circ g$ (it's called Faa di Bruno's formula but the formula itself is not my question). In their very ...
3
votes
1answer
33 views

“Reduction of Dirichlet series into power series”

In a paper of Riemann, he states to following formal identity. If $f(s)=\sum\limits_{k=1}^{\infty}\frac{a_k}{k^s}$ and $F(x)=\sum\limits_{k=1}^{\infty}a_kx^k$ then ...
0
votes
1answer
106 views

If $f(2x)=2xf'(x)$, then find $f(x)$

If $f(x)$ is Analytic functions on $R$,and such $$2xf'(x)=f(2x)$$ Find all $f(x)$ My idea: let $$f(x)=\sum_{n=0}^{\infty}a_{n}x^n$$ so I can't Thank you