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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42
votes
11answers
4k 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 ...
2
votes
3answers
537 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 ...
6
votes
4answers
1k 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
4answers
893 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 ...
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 ...
13
votes
2answers
313 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
562 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?
5
votes
2answers
666 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
771 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 ...
3
votes
4answers
409 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
205 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 $0$. ...
3
votes
3answers
354 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 ...
12
votes
4answers
867 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?
12
votes
2answers
537 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 $\{a_k\}\subset\mathbb{N}$ ...
9
votes
1answer
150 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 ...
5
votes
3answers
233 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
1answer
145 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
3answers
349 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?
3
votes
2answers
130 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 ...
1
vote
2answers
189 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
249 views

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

Consider the following 2D infinitely large grid where the dots represent infinity: ...
42
votes
1answer
848 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, ...
34
votes
3answers
2k 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 ...
23
votes
2answers
1k 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) ...
21
votes
1answer
794 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
850 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
4answers
301 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?
11
votes
1answer
459 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 ...
10
votes
3answers
254 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
255 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 = ...
3
votes
2answers
91 views

Power series relation

Draks gave the identity, Higher Order Trigonometric Function $$\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}=\frac{1}{m}\sum_{k=0}^{m-1} \exp( e^{i\frac{2k+1}{m}\pi}x )$$ How can this be proven?
3
votes
2answers
382 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 ...
11
votes
4answers
601 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 ...
3
votes
3answers
187 views

Asymptotic expansion of $J(t) = \int^{\infty}_{0}{\exp(-t(x + 4/(x+1)))}\, dx$

I want to derive an asymptotic expansion for the following Bessel function. I think I need to rewrite it in another form, from which I can integrate it by parts. I am interested in obtaining the ...
3
votes
3answers
232 views

the sum of a series

I am stuck on the computation of the following sum: $$\sum_{k=0}^{\infty} {\Big( {\frac{q}{k+1}} \Big)}^k ,$$ where $k$ is a natural number, and $0<q<1$.
9
votes
3answers
199 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
3answers
248 views

Can the the radius of convergence increase due to composition of two power series?

When composing power series, is the radius of convergence the minimum of that of the individual series, or is it like for multiplication and addition of power series where the resultant radius of ...
7
votes
3answers
530 views

The Power of Taylor Series

I am teaching a Calculus class and we are finishing up power/Taylor series this week. The last section of the chapter is on applications, but the only ones listed there are approximating non-rational ...
7
votes
2answers
756 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 ...
6
votes
1answer
165 views

Existence of a power series converging non-uniformly to a continuous function

I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that: $f$ converges and is continuous on the closed unit disk $D$ and the series $\sum_n a_n z^n$ does not converge ...
6
votes
3answers
341 views

Expressing the area of the image of a holomorphic function by the coefficients of its expansion

I have the following problem. Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set ...
6
votes
4answers
353 views

Power Series with the coefficients $n!/(n^n)$

I'd be grateful if someone could tell me how to obtain the convergence radius of the aforementioned power series. Or, by Cauchy Hadamard, the limit of $(n!/(n^n))^{(1/n)}$ as n approaches infinity. ...
5
votes
2answers
109 views

Sum up to number $N$ using $1,2$ and $3$

So the question asked was finding out the number of ways(combinations), a given number $N$ can be formed using the sum of $1,2$ or $3$. (eg) For $n = 8$, the answer is $10$ The given solution for ...
3
votes
1answer
240 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
1answer
345 views

Prove that if $\sum a_k z_1^k$ converges, then $\sum a_k z^k$ also converges, for $|z|<|z_1|$

Show that if a power-series converges for any value of $z_{0}$ of $z$, it will be absolutely convergent for all values of $z$ whose representation points are within a circle which passes through ...
1
vote
2answers
117 views

Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$ [duplicate]

Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$. I have no idea to solve this problem. Anyone could help me?
4
votes
3answers
426 views

What would be the radius of convergence of $\sum\limits_{n=0}^{\infty} z^{3^n}$?

I know how to find the radius of convergence of a power series $\sum\limits_{n=0}^{\infty} a_nz^n$, but how does this apply to the power series $\sum\limits_{n=0}^{\infty} z^{3^n}$? Would the ...
3
votes
0answers
71 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
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 ...