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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0answers
44 views

Borel-/Laplace-transform and $\psi$-function

I'm considering some family of functions whose coefficients of their power series occur in the columns of the following matrix A (of course thought as of infinite size) $ \qquad $ The ...
4
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2answers
64 views

How to calculate what this power series converges against? (double factorials)

I'm working on my physics master course homework and I'm given the following equation out of nowhere: $\displaystyle{ 1 + \sum_{n\ =\ 1}^{\infty}{z^n\left(\, 2n - 1\,\right)!! \over 2n!!} ={1 \over ...
0
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2answers
34 views

Prove/disprove that $\sum_0^\infty a_nx^n = 0 \rightarrow a_n = 0 \text{ for all }n$ given $|x|<1$

For $|x|<1$, if the following statement is true, how to prove it? If not, how to disprove it? $$\sum_0^\infty a_nx^n = 0 \rightarrow a_n = 0 \text{ for all }n$$ In case $x$ takes any real value, ...
1
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1answer
85 views

Proof that $e^x$ can be expressed in a series of ascending powers of $x$

In a pure maths textbook I have, they prove that $e^x$ can be expressed as $1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\frac{x^4}{4!}+\ldots+\frac{x^n}{n!}+\ldots$ However, before they prove this, they say they ...
0
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1answer
12 views

Right derivative of a power series

I have a power series $\displaystyle \sum \limits_{n \in \mathbb{N}} a_{n}x^{n}$ whose radius of convergence is equal to $4$. For all $x \in ]-4,4[$, let $f(x) = \displaystyle \sum_{n=0}^{+\infty} ...
0
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0answers
12 views

When generating function solution is valid analytical solution

When can I assume that closed form acquired by methods used for generating functions is valid also for analytical power series. In other words if there is closed form for convergent power series will ...
1
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0answers
29 views

Multiplication of a complex function with essential singularity with another complex function with a pole at the same point

Im trying to proof or disprove the following claim: If $f(z)$ and $g(z)$ are holomorphic in an annulus $0 < |z − z(\beta)| < R$ and $f$ has an essential singularity at $z(\beta)$ and $g$ has a ...
2
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2answers
70 views

Function such that $f^{(n)}(0)=\frac{(n!)^2}{(2n+1)!}$

I was trying to solve another problem and come up with the problem if there is a function with closed form such that $$f^{(n)}(0)=\frac{(n!)^2}{(2n+1)!};(n\ge1).$$ I tried to check the condition for ...
0
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1answer
31 views

Sum of the series $\sum_{i=1}^n a^i i^r$

How can I find the sum of the series : $$\sum_{i=1}^n a^ii^r $$
3
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2answers
62 views

A simple Partial Differential Equation? I thought…

I was looking at the Partial Differential Equation involving function: $$ z(x,y)$$ $$ \frac{\partial z}{\partial x} + c \frac{\partial z}{\partial y} = 0 $$ Which fairly intuitively has a ...
2
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0answers
28 views

Finding the First Few Terms of the Sum of Two Infinite Series

Consider $$f(x)=\sum_{n=0}^{\infty}\sum_{j=0}^{n}\sum_{k=0}^{j}d_{k}c_{j-k}a_{n-j}x^{n-11}+\sum_{n=0}^{\infty}\sum_{k=0}^{n}e_{k}b_{n-k}x^{n-8}.$$ Given that I know the first few values of ...
2
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1answer
39 views

Complex Power Series Weird Convergence

I'm trying to find a compex power series centered at $i$ with convergence radius $\sqrt{2}$ which converges for $z=1$ but not for $z=-1$. Any help?
0
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2answers
18 views

Why limes superior in Cauchy-Hadamard formula for radius of convergence of power series?

Can anyone explain to me why there is $\limsup$ instead of $\lim$ in Cauchy-Hadamard formula for radius of convergence of power series? It isn't that obvious to me ;/
3
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0answers
106 views

Is there a way to exploit local redundancy in a function to speed up Monte Carlo integration?

In every Monte Carlo method I've ever seen, $f$ must be recomputed from scratch for each point that is (somehow randomly) selected to contribute to the overall integral. However, most functions have ...
0
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0answers
32 views

Find present value of employee given human capital investment

Here is the problem that I wanted to get a directions on: John Doe earns $80000$ a year. By 6th month of his employment company's HR determined that John needs human capital investment totaling ...
3
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2answers
38 views

help on a specific power series expansion, i cannot see what the author did here

I am working through the next chapter of my quantum mechanics book over winter break, and admittedly series are my weakest point as far as calculus is concerned. The author starts with the ...
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0answers
46 views

Why does $a^m = \sum_0^\infty a^i$?

It is stated in in this comment that: $$(x^{n_1}+ x^{n_2}+\cdots+x^{n_k})^m = \sum_{n=0}^{\infty} F(n;n_1,\cdots,n_k) x^n = \\ = 1+(x^{n_1}+ x^{n_2}+\cdots+x^{n_k}) + (x^{n_1}+ ...
0
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3answers
80 views

If we want to write a function in power series, what's the first n? 0 or 1?

So, for instance $e^x = \sum\limits_{n=0}^\infty \frac{x^n}{n!}$, why do we start from n=0, and not n=1? Why do we care about some zero$^{th}$ term? Do we do the same with other functions? Why?
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0answers
36 views

borel summability- sum of power series

let (x_k) be any convergence sequence. and let (x_k) converges to c in R. Then show that the power series B=e^(-n) Σ(x_k)((n^k)/k!) ...(the indices are k=0 to infinity.)I am trying to find the ...
1
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1answer
24 views

Functional equation for polynomials

While reading a chapter entitled "Functional equations for polynomials" in the book "Polynomials" by Victor Prasolov, he states that Every polynomial $f$ of degree $n+1$ satisfies the identity ...
5
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4answers
116 views

Show that all derivatives of $f(x) = \frac{\sin(2\pi x)}{\pi x}$ belong to $L^2(\mathbb{R})$

I would like to show that all derivatives of $f(x) = \frac{\sin(2\pi x)}{\pi x}$ belong to $L^2(\mathbb{R}).$ One route I have tried is a power series approach. If we expand our function $f$ in a ...
0
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1answer
38 views

Simplify Infinite Series Involving Gamma Function $\Gamma$

This question originated from this problem. Can anyone help me simplifying the infinite series below: $$\sum_{n=0}^{+\infty}\frac{1}{\Gamma(\beta_2-n)}\frac{(-e^{-t})^n}{n!}$$ The only idea I have ...
0
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2answers
55 views

Show that two power series have the same radius of convergence

Show that the power series $\sum_{n=0}^{\infty}{c_{n}x^{n}}$ has the same radius of convergence as $\sum_{n=0}^{\infty}{c_{n+m}x^{n}}$ for any positive integer $m$. Should I use the Ratio Test? Or do ...
4
votes
3answers
152 views

Closed form for series $\sum_{m=1}^{N}m^n\binom{N}{m}$ [duplicate]

How can we calculate the series $$ I_N(n)=\sum_{m=1}^{N}m^n\binom{N}{m}? $$ with $n,N$ are integers. The first three ones are $$ I_N(1)=N2^{N-1}; I_N(2)=N(N+1)2^{N-2}; I_N(3)=N^2(N+3)2^{N-3} $$
4
votes
1answer
55 views

Convergence of power series with eventually constant coeffcients

Assume I have a sequence $f_n$ of power series of the form $$ f_n(x) = \sum_{i=0}^\infty{a_{n,i}x^i},\quad a_{n,i}=\begin{cases}\alpha_{n,i} & n\leq i,\\b_i & n>i.\end{cases}.\tag{*} $$ ...
0
votes
1answer
49 views

Find the radius of convergence of the Taylor series of $f(x) = \frac{x-3}{x+2}\ln(5+x)$ at $x=0$

I need find radius of convergence for Taylor series in $x = 0$ (over $\mathbb{R}$) and find $x$'s at which series converges to $f$ $$f(x) = \frac{x-3}{x+2}\ln(5+x)$$ My solution $\ln(5+x) = ...
1
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5answers
162 views

Power series for the rational function $(1+x)^3/(1-x)^3$

Show that $$\dfrac{(1+x)^3}{(1-x)^3} =1 + \displaystyle\sum_{n=1}^{\infty} (4n^2+2)x^n$$ I tried with the partial frationaising the expression that gives me $\dfrac{-6}{(x-1)} - ...
0
votes
1answer
58 views

What is the radius of convergence of the power series $\sum_{n=0}^\infty \frac {(n!)^k}{(kn)!}z^n$?

What is the radius of convergence of the power series? $$\sum_{n=0}^\infty \frac {(n!)^k}{(kn)!}z^n$$ Progress Used the ratio test, but got $0$ from it.
3
votes
1answer
29 views

Power series for inverse of truncated power series of $e$

Let $T_n(x)=\sum_{k=0}^{n}\frac{x^k}{k!}$. I'm looking at the function $$f(x)=\frac{1}{T_n(x)}$$ and I would like to find the power series of this particular function. I know I can use Maclaurin ...
2
votes
1answer
40 views

Powers and Power Series'

When doing some problems I came across the function: $$f(x)=\frac{x}{1-2x}$$ I realised that the Maclaurin expansion of this function was: $$f(x)=x+2x^2+4x^3+16x^4...$$ Evaluate at $x=1$ to get ...
-1
votes
4answers
47 views

Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$?

Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$? I know $\lim_{n \to \infty} \sum_{k=0}^n\frac{t^k}{k!}=e^t$, but my sum starts at $k=1$ and also has ...
4
votes
0answers
88 views

Calculate $1^1 + 2^2 + 3^3 + … + n^n$

Is there a formula to calculate $1^1 + 2^2 + 3^3 + ... + n^n$ I searched but didn't find a formula for increasing powers
0
votes
0answers
45 views

Truncation of partitions generating function question

$A (x)$ is the generating function for partitions. $B(x)=\sum_{n=0}^{\infty}b_nx^n $ $$b_n =\binom{\text{number of partitions of }n}{\text{into an even number of parts}}-\binom{\text{number of ...
1
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3answers
96 views

Series expansion of $\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$?

How would I find the series expansion $\displaystyle\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$ so that it will turn into an infinite power series again??
1
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2answers
87 views

$\int_{-L}^{+L}h(z)\,dz = 2 \sum_{-\infty}^{+\infty}\frac {a_n}{n} \sin (nL)$

is it possible to find a formula for $a_n$ from $$\int_{-L}^{+L}h(z)\,dz = 2 \sum_{-\infty}^{+\infty}\frac {a_n}{n} \sin (nL)$$ For $n=0$ the series is $0$ Thanks
4
votes
1answer
101 views

How to write $1-x-x^3+x^4+x^5+x^6-x^7 \cdots$ as a power series representation

How can I write $1-x-x^3+x^4+x^5+x^6-x^7 ....$ as a power series representation (i.e., a neat fraction such as $\frac{1}{1-x}$. This stems from $\binom{\text{number of partitions of }n}{\text{into an ...
2
votes
3answers
71 views

Writing $1+3x^2+8x^4+21x^6+\cdots$ as a power series representation

How would I write the power series $$1+3x^2+8x^4+21x^6+\cdots$$ as a power series representation (something neat similar to $\frac{1}{1-x}$)? This reminds me of the power series ...
5
votes
4answers
299 views

Find the fraction where the decimal expansion is infinite?

Find the fraction with integers for the numerator and denominator, where the decimal expansion is $0.11235.....$ The numerator and denominator must be less than $100$. Find the fraction. I ...
0
votes
1answer
46 views

Using the Maclaurin series to approximate $f(0.1)$ for $f(x)=(3+e^{2x})^{0.5}$

I was tasked to use the Maclaurin series to calculate $f(0.1)$ of $f(x)=(3+e^{2x})^{0.5}$. I got the Maclaurin expansion of $p_2(x) = \sqrt{3} + 4x +5x^2$ into which I plugged $0.1$ to yield ...
4
votes
1answer
93 views

Adding Two Power Series if their bounds are different

I have the following product $$\frac{1}{6}\sum_{n=0}^{\infty}\left(\frac{-x}{2}\right)^n\sum_{n=0}^{\infty}\frac{a_nx^{n+2}}{n!}$$ Where $a_n$ is an arbitrary coefficient. If I factor out $x^2$ ...
6
votes
4answers
131 views

How can I write this power series as a power series representation?

How can I write this power series ($1+x+2x^2+2x^3+3x^4+3x^5+4x^6+4x^7+5x^8....$) as a power series representation (like $\dfrac{1}{1-x}$ or something neat like that)?
2
votes
1answer
44 views

Determining a radius convergence of a power series

Let $$ \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1} $$ Is there an immediate way to determine $R=1$?
3
votes
1answer
47 views

Radius of convergence, prove that $\sum\limits_{n=0}^{\infty} a_n z^n$ converges absolutely for every $z \in \mathbb{C}$ with $|z| < R$

A proof of this is given in my lecture notes as follows: We define $R$ to be $\sup \{|z| \in \mathbb{R} : \sum |c_k z^k|$ converges $\}$ when the supremum exists. Prove that $\sum |c_k ...
3
votes
2answers
231 views

How to calculate the series?

How can we calculate the series: $$ F(x)=\sum_{n=1}^{\infty}\frac{(-1)^n}{1-x^n} $$ I found that $$ ...
2
votes
1answer
40 views

Find the leftmost (most significant digits) of a large exponent calculation, say $99^{99}$

I want to find the initial 10 digits of an exponent calculation whose result is a very large number - Say, $99^{99} = 3.697296 \times 10^{197}$ I only need to know the digits $3697296$ Is there any ...
1
vote
0answers
35 views

derivative of t distribution cdf wrt degrees of freedom

Given the cdf of a t distribution as follows: $T_\nu(x)=\frac{1}{2} + x\Gamma(\frac{\nu+1}{2}) + \frac{_2F_1 ...
1
vote
1answer
54 views

Can this infinite summation be simplified?

I encountered the following infinite summation $$\sum_{k=0,k\neq m}^{\infty}\frac{x^k}{(k-m)k!},x>0,$$ can it be simplified? Thanks!
3
votes
2answers
69 views

Prove two identities relating to series

Show that: $$ (1). \sum_{n=1}^{\infty}\ln\big(\cos \frac{x}{2^n}\big)=\ln \frac{\sin x}{x} $$ $$ (2). \sum_{n=1}^{\infty}\frac{1}{2^n}\tan \frac{x}{2^n}=\frac{1}{x}-\cot x $$ Thank you in advance. ...
0
votes
4answers
48 views

Infinite Sum of 1/Polynomial

I'm trying to solve this equation: $$\sum_{k = 0}^{\infty}\dfrac{1}{(k+1)(k+3)}$$ Original image at http://i.imgur.com/wXZFxn0.png I attempted to find the sums of $\sum_0^∞\frac{1}{k+1}$ and ...
1
vote
2answers
69 views

Questionable Power Series for $1/x$ about $x=0$

WolframAlpha states that The power series for $1/x$ about $x=0$ is: $$1/x = \sum_{n=0}^{\infty} (-1)^n(x-1)^n$$ This is supposedly incorrect, isnt it? This is showing the power series about ...