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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195
votes
17answers
14k views

How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
11
votes
3answers
2k views

Product of two power series

Say if I define a power series over some arbitrary field $F$ as $$a = \sum^{ \infty }_{i = 0} a_{i} X^{i} $$ Then can I say: $$ab = \sum^{ \infty }_{i = 0} \sum^{ \infty }_{j = 0} a_{i} b_{j} X^{i ...
5
votes
2answers
2k views

About the limit of the coefficient ratio for a power series over complex numbers

This is my first question in mathSE, hope that it is suitable here! I'm currently self-studying complex analysis using the book by Stein & Shakarchi, and this is one of the exercises (p.67, Q14) ...
7
votes
2answers
2k views

Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$

How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in ...
2
votes
3answers
2k views

Difficulties performing Laurent Series expansions to determine Residues

The following problems are from Brown and Churchill's Complex Variables, 8ed. From §71 concerning Residues and Poles, problem #1d: Determine the residue at $z = 0$ of the function $$\frac{\cot(z)}{...
12
votes
4answers
1k views

Summing the power series $\sum\limits_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1}\prod\limits_{k=1}^n\frac{2k-1}{2k} $

I'd like to determine the function corresponding to the following power series: $$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1}, $$ where $|...
2
votes
2answers
6k views

Sum of a power series $n x^n$ [duplicate]

I would like to know: How come that $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$ Why isn't it infinity?
3
votes
4answers
1k views

Formula for calculating $\sum_{n=0}^{m}nr^n$

I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived. For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ ...
2
votes
3answers
692 views

Find the form of a second linear independent solution when the two roots of indicial equation are different by a integer

Consider the differential equation $$x^2y''+3(x-x^2)y'-3y=0$$ $(a)$ Find the recurrence equation and first three nonzero terms of the series solution in powers of $$ corresponding to the larger root ...
6
votes
4answers
2k views

Formal power series coefficient multiplication

Given that I have two formal power series: $$ A(x) = \sum_{k \ge 0} a_k x^k $$ $$ B(x) = \sum_{k \ge 0} b_k x^k $$ The Cauchy Product gives a series $$ C(x) = \sum_{k \ge 0} c_k x^k $$ $$ c_k = \...
10
votes
3answers
781 views

How to calculate $f(x)$ in $f(f(x)) = e^x$?

How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$? Is there a faster-converging method than power series for fractional iteration/functional square roots?
16
votes
3answers
764 views

What is the Riemann surface of $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$?

The following appears as the second-to-last problem of Stewart's Complex Analysis: Describe the Riemann surface of the function $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$. This problem ...
11
votes
1answer
1k views

Deriving Maclaurin series for $\frac{\arcsin x}{\sqrt{1-x^2}}$.

Intrigued by this brilliant answer from Ron Gordon, I was attempting to find the Maclaurin series for $$f(x)=\frac{\arcsin x}{\sqrt{1-x^2}}=g(x)G(x)$$ with $g(x)=\frac{1}{\sqrt{1-x^2}}$ and $G(x)$ ...
4
votes
2answers
178 views

How to find the radius of convergence?

The function is $\dfrac {z-z^3}{\sin {\pi z}} $. How to find the radius of convergence in $ z=0 $?
3
votes
5answers
368 views

Identify infinite sum: $\sum\limits_{n=0}^{+\infty}\frac{x^{4n}}{(4n)!}$

Find $f(x)$, the unknown function satisfying $$f(x) = \sum\limits_{n=0}^{+\infty}\frac{x^{4n}}{(4n)!}$$ I'm looking for a direct solution which is different from mine, if possible.
3
votes
2answers
4k views

Radius of convergence of a sum of power series

I have two series $\displaystyle\sum_{n=1}^{\infty} a_n x^{n}$ $\displaystyle\sum_{n=1}^{\infty} b_n x^{n}$ with radius of convergence $2$ and $3$ respectively. How can I find the radius of ...
32
votes
2answers
2k views

Proof that $\sum\limits_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$ regarding $\zeta(3)$ and Apéry's proof

I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$ $$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$ and thus (probably) ...
14
votes
1answer
502 views

Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$?

This was inspired by this post. Let $q = e^{2\pi\,i\tau}$. Then, $$x := \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$ ...
10
votes
4answers
170 views

Bernoulli Number analog using Cosine

I know that Bernoulli Numbers can be found with the generating function $$\frac{x}{e^x-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n$$ I was wondering if any work has been done using a similar equation $$\...
3
votes
1answer
635 views

Riemann Zeta Function Manipulation

The Riemann zeta function is defined on the $Re z> 1$ by $$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}$$ (i) show that for $Re z> 1$, we have $$(1-2^{1-z})\zeta(z)=\sum_{n=1}^\infty \frac{(-1)^{n+...
4
votes
1answer
244 views

Uniform convergence in the endpoints of an interval

Study the pointwise and uniform convergence of the series $$\sum_{n=1}^\infty\dfrac{4^n}{n^2}\dfrac1{(1+x^2)^n}$$ I'm doing this exercise and I'm not sure about the following: What I've done is ...
0
votes
1answer
188 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
5
votes
3answers
260 views

Summing up the series $a_{3k}$ where $\log(1-x+x^2) = \sum a_k x^k$

If $\ln(1-x+x^2) = a_1x+a_2x^2 + \cdots \text{ then } a_3+a_6+a_9+a_{12} + \cdots = $ ? My approach is to write $1-x+x^2 = \frac{1+x^3}{1+x}$ then expanding the respective logarithms,I got a series (...
31
votes
1answer
1k views

Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius?

It is well-known that the function $$f(x) = \begin{cases} e^{-1/x^2}, \mbox{if } x \ne 0 \\ 0, \mbox{if } x = 0\end{cases}$$ is smooth everywhere, yet not analytic at $x = 0$. In particular, its ...
14
votes
4answers
1k views

How to express $(1+x+x^2+\cdots+x^m)^n$ as a power series?

Is it possible to express $(1+x+x^2+\cdots+x^m)^n$ as a power series?
13
votes
2answers
657 views

How to do a very long division: continued fraction for tan

I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in http://arxiv.org/...
8
votes
1answer
148 views

Proving that $\lim_{x\to1^-}\left(\sqrt[a]{1-x}\cdot\sum_{n=0}^\infty~x^{n^a}\right)=\Gamma\left(1+\frac1a\right)$

How could we prove that $$\lim_{x\to1^-}~\bigg(\sqrt[a]{1-x}\cdot\sum_{n=0}^\infty~x^{n^a}\bigg)~=~\Gamma\bigg(1+\frac1a\bigg)$$ for $a>0$ ? The inspiration came to me while trying find a ...
3
votes
3answers
707 views

Cauchy product on exponential-looking power series

Original posting by dioxen here: Double summation including power and factorial I am finding some trouble in computing the following sum: $$\sum_{k=0}^\infty \frac{x^k}{k!}\;\sum_{m=0}^k\frac ...
3
votes
2answers
510 views

Series of nested integrals

I'm trying to calculate the following series of nested integrals with $\varepsilon(t)$ being a real function. $$\sigma = 1 + \int\nolimits_{t_0}^t\mathrm dt_1 \, \varepsilon(t_1) + \int_{t_0}^t\...
11
votes
1answer
801 views

Radius of convergence of power series

Given a meromorphic function on $\mathbb{C}$, is the radius of convergence in a regular point exactly the distance to the closest pole? As Robert Israel points out in his answer, that this is of ...
6
votes
2answers
439 views

The value of a limit of a power series: $\lim\limits_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$

What is the answer to the following limit of a power series? $$\lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$$
5
votes
3answers
236 views

Series of inverses of binomial coefficients

Can you think of a simple way of proving that $$ \sum_{n=k+1}^\infty \frac{1}{n \choose k} $$ is rational for any $k \geq 2$? Here's the background. Consider a series: $$ \sum_{n=1}^\infty \frac{1}{...
12
votes
2answers
631 views

$\lim\limits_{x\to\infty}f(x)^{1/x}$ where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$.

Does the following limit exist? What is the value of it if it exists? $$\lim\limits_{x\to\infty}f(x)^{1/x}$$ where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$ and $\{a_k\}\subset\mathbb{N}$ ...
7
votes
6answers
505 views

Why is $ \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x$?

I am trying to see where this relationship comes from: $\displaystyle \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x$ Does anyone have any special knowledge that me and my summer math teacher doesn't know ...
5
votes
1answer
245 views

Why don't we have an isomorphism between $R[x]$ and $ R[[x]]$?

$R$ is a ring. Why don't we have an isomorphism $$ R[x] \cong R[[x]]\ ? $$
4
votes
3answers
1k views

Taylor series for different points… how do they look?

I can't understand what it means to do the Taylor series at the point $a$. The best way would be showing me how it looks for different $a$ on a graph. Do I find those graphs on the Internet?
12
votes
3answers
242 views

Could we show $1-(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dots)^2=(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}- \dots)^2$ if we didn't know about Taylor Expansion?

Suppose that humanity haven't discovered Taylor Series Expansion of trigonometric functions or of any function that would help us on this. Which means we are not allowed to replace the given infinite ...
9
votes
1answer
203 views

How to calculate the integral of $x^x$ between $0$ and $1$ using series? [duplicate]

How to calculate $\int_0^1 x^x\,dx$ using series? I read from a book that $$\int_0^1 x^x\,dx = 1-\frac{1}{2^2}+\frac{1}{3^3}+\dots+(-1)^n\frac{1}{(n+1)^{n+1}}+\cdots$$ but I can't prove it. Thanks in ...
7
votes
2answers
1k views

What is the radius of convergence of $\sum z^{n!}$?

How to find the radius of convergence of $\sum z^{n!}$? I'm used to applying the ratio test to power series of the form $\sum a_{n}z^{n}$, but for a different power of $z$, I am a bit stumped. What ...
4
votes
1answer
200 views

Is there a generalization of the fundamental theorem of algebra for power series?

Given the similarity between polynomials and power series, I was wondering if there is any generalization of the fundamental theorem of algebra for power series. I understand that it doesn't make much ...
2
votes
1answer
173 views

Prove the following equation of complex power series.

Show that for $|z| \lt 1$ with $z \in \Bbb C$, we have $$ \sum_0^\infty \frac{{z^2}^k}{1-{z^2}^{k+1}} = \frac{z}{1-z} $$ $$ \sum_0^\infty \frac{2^k{z^2}^k}{1+{z^2}^{k}} = \frac{z}{1-z} $$ My guess ...
2
votes
5answers
1k views

How to create alternating series with happening every two terms

I'm looking for a technique for creating alternating negatives and positives in a series. Specifically: when n=1, the answer is +, n=2 is +, n=3 is -, n=4 is -... etc. I have every other part of the ...
3
votes
1answer
155 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 $\sum\limits_{k=0}^{n}...
3
votes
1answer
119 views

Proving the inequality $|e^z-1|\leq e^{|z|}-1$

I am trying to prove this inequality $$|e^z-1|\leq e^{|z|}-1\leq |z|e^{|z|}$$ I've tried calculating the difference in their power series $$\sum_{k=0}^\infty\frac{|z|^k}{k!}-1-\left|\,\sum_{k=0}^\...
2
votes
3answers
232 views

Evaluate this power series

Evaluate the sum $$x+\frac{2}{3}x^3+\frac{2}{3}\cdot\frac{4}{5}x^5+\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}x^7+\dots$$ Totally no idea. I think this series may related to the $\sin x$ series ...
1
vote
3answers
92 views

Compute the sum $\sum_{k=1}^{\infty}k^mz^k$ where $|z|<1$

Do you know how to find the limit of $\sum_{k=1}^{\infty}k^mz^k$ where $|z|<1$ and m is a natural number? I've tried to google it in wiki but I do not understand the closed form (http://en....
1
vote
2answers
453 views

How to construct this Laurent series?

How do I construct the following Laurent series (clipped off Wolfram Alpha)? I know that the numerator can be written as $-1+\frac{\pi}2 z-...$ Alternatively (without the Laurent series), how can I ...
1
vote
3answers
113 views

Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation

Find sum of the series $$\sum_{n=1}^{\infty}\frac{x^{3n}}{\left(3n\right)!}$$ using differentiation. So far I found that $$S(x)+1=S'''(x)$$ but it does not help. Then I tried different interesting ...
1
vote
1answer
55 views

Does the Abel sum 1 - 1 + 1 - 1 + … = 1/2 imply $\eta(0)=1/2$?

If $\sum_{n=1}^\infty a_n$ is Abel summable to $A$, then necessarily $\sum_{n=1}^\infty a_n n^{-s}$ has a finite abscissa of convergence and can be analytically continued to a function $F(s)$ on a ...
61
votes
4answers
5k views

Factorial and exponential dual identities

There are two identities that have a seemingly dual correspondence: $$e^x = \sum_{n\ge0} {x^n\over n!}$$ and $$n! = \int_0^{\infty} {x^n\over e^x}\ dx.$$ Is there anything to this comparison? (I ...