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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131
votes
15answers
9k 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 ...
5
votes
2answers
1k 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) ...
5
votes
2answers
1k 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
1k 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 ...
2
votes
3answers
415 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 ...
10
votes
4answers
1k views

Help with summing a power series

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 ...
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 = ...
4
votes
2answers
140 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 $?
11
votes
3answers
1k 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 ...
3
votes
4answers
799 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
5answers
322 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.
30
votes
2answers
2k views

Proving an “amazing” claim 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) ...
13
votes
2answers
516 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 ...
8
votes
3answers
707 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?
11
votes
1answer
616 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 ...
9
votes
4answers
141 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 ...
8
votes
1answer
801 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 $G(x)$ ...
5
votes
3answers
251 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 ...
2
votes
3answers
211 views

What is the expression for this summation?

Known that $\sum_{n=0}^{\infty}{x^n}{z^n}=\frac{1}{1-xz}$. If we have $\sum_{n=0}^{\infty}\frac{{x^n}{z^n}}{n\beta + \alpha}$ where $\beta, \alpha $ are element of real numbers but not equal $0$. ...
1
vote
2answers
2k 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?
16
votes
3answers
423 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 ...
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?
3
votes
3answers
560 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
1answer
438 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 ...
3
votes
2answers
451 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_{t_0}^t\mathrm dt_1 \, \varepsilon(t_1) + \int_{t_0}^t\mathrm dt_1 ...
12
votes
2answers
614 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 $\{a_k\}\subset\mathbb{N}$ ...
3
votes
3answers
713 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?
9
votes
1answer
169 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 $\displaystyle\sum z^{n!}$?

How could you find out the radius of convergence of $\displaystyle\sum z^{n!}$? I'm used to applying the ratio test to power series of the form $\displaystyle\sum a_{n}z^{n}$, but for a different ...
2
votes
2answers
645 views

Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$ [closed]

Show that $$\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)},$$ where $0\leq r <1$. Using this, prove that $\sum_{n=0}^\infty r^n ...
2
votes
1answer
163 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 ...
0
votes
1answer
101 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 ...
3
votes
1answer
138 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 ...
3
votes
2answers
135 views

How to simplify $f(x)=\sum\limits_{i=0}^{\infty}\frac{x^{i \;\bmod (k-1)}}{i!}$?

$$f(x)=\sum_{i=0}^{\infty}\frac{x^{i \;\bmod (k-1)}}{i!}$$ ${i \bmod (k-1)}$ $\quad$ says the $x$ powers can be only $x^0$, $x^1$, ...,$x^{k-2}$ Understand simplify a way to transform this infinity ...
2
votes
3answers
198 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
77 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 ...
1
vote
2answers
376 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 ...
0
votes
1answer
273 views

Calculating powers of 2 on a 2D grid without factoring.

Consider the following 2D infinitely large grid where the dots represent infinity: ...
46
votes
3answers
3k 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 ...
47
votes
1answer
1k views

Fractal behavior along the boundary of convergence?

The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, ...
24
votes
1answer
822 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 ...
28
votes
2answers
817 views

How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.

Prove that there is no continuous functions $f:[0,1]\to R$, such that $$ f(x)+f(x^2)=x. $$ My try. Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ and all ...
21
votes
1answer
938 views

How to prove convergence of polynomials in $e$ (Euler's number)

These polynomials in $e$ converge to 2$$f(i)=e^i - i \sum_{k=1}^{i-1}\frac{(i-k)^{k-1}{e^{i-k}}{(-1)^{k+1}}}{k!}, \text{ where } i>1$$ This function goes to 2. I've calculated this with sage math ...
21
votes
5answers
1k views

Sum of the form $r+r^2+r^4+\dots+r^{2^k} = \sum_{i=1}^k r^{2^k}$

I am wondering if there exists any formula for the following power series : $$S = r + r^2 + r^4 + r^8 + r^{16} + r^{32} + ...... + r^{2^k}$$ Is there any way to calculate the sum of above series (if ...
14
votes
2answers
1k views

any pattern here ? (revised 2)

for any positive number $k$, I have a $(k+1)*(k+1)$ matrix. I wonder if these matrices follow any "obvious" pattern. My goal is to guess the elements for matrix with $k=5$ and above (most probably in ...
5
votes
5answers
321 views

prove that $(1 + x)^\frac{1}{b}$ is a formal power series

How to prove that $(1 + x)^\frac{1}{b}$ (where $b$ is an integer) can be expressed as a formal power series without using Binomial theorem?
13
votes
2answers
816 views

Ramanujan's approximation for $\pi$

In 1910, Srinivasa Ramanujan found several rapidly converging infinite series of $\pi$, such as $$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}. ...
12
votes
3answers
980 views

Taylor expansion of $(1+x)^α$ to binomial series – why does the remainder term converge?

For $α ∈ ℝ$ the function $g_α \colon B_1(0) → ℝ, x ↦ (1+x)^α$ is $C^∞$ and $g_α^{(n)}(x) = n! \tbinom{α}{n}(1+x)^{α-n}$, where $\tbinom{α}{n} = \frac{α(α-1)\cdots(α-n+1)}{n!}$ is the generalized ...
10
votes
3answers
296 views

Estimate $\displaystyle\int_0^\infty\frac{t^n}{n!}e^{-e^t}dt$ accurately.

How can I obtain good asymptotics for $$\gamma_n=\displaystyle\int_0^\infty\frac{t^n}{n!}e^{-e^t}dt\text{ ? }$$ [This has been already done] In particular, I would like to obtain asymptotics that ...
10
votes
1answer
439 views

Abel limit theorem

I would like to know if the Abel limit theorem works if the limit is infinite. Let the series $\sum_{k=0}^\infty a_k x^k$ have radius of convergence 1. Assume further that $\sum_{k=0}^\infty a_k = ...