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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1
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1answer
342 views

Fibonacci Generating Function of a Complex Variable

So I'm doing work on the Fibonacci Numbers, and I came across this problem for the generating function for the recursive fibonacci numbers. I have two questions: 1. Why is it useful to use a ...
1
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1answer
43 views

Prove that $f^{(k)}(0) = \frac{k!}{2πi} \int_{|z|=1} \frac{f(z)}{z^{k+1}} dz$

Let $f(z)$ be a convergent power series with convergent radius greater than 1. Prove that $$f^{(k)}(0) = \frac{k!}{2πi} \int_{|z|=1} \frac{f(z)}{z^{k+1}} dz$$ Since $f(z)$ is a convergent power ...
1
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0answers
76 views

Asymptotic Expansion for a Function involving a Weird Integral

So I'm trying to find the asymptotic expansion as $x \to \infty$ of $$f(x)=\frac{1}{\bigg[A-\int \frac{\lambda^x}{\Gamma(x+1)}dx\bigg]^\frac{1}{\alpha}}$$ Note that $\lambda>0$ and $\alpha>0$. ...
0
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3answers
57 views

A simple series

I don't do math a long time, so I completely don't remember how to prove that: $$ \sum_{i=1}^\infty \frac{i}{2^i} = 2 $$ Can anybody help me?
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2answers
88 views

Showing that $f(x)^3 + g(x)^3 + h(x)^3 - 3f(x)g(x)h(x) = 1$ for functions $f$, $g$, and $h$ defined by certain power series [duplicate]

I'm having trouble with this question, I have found the interval of convergence of $h(x)$ to be $(-\infty, \infty)$, but I don't know how to use that for the question as well as the hint. Any help ...
0
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2answers
129 views

How to find $a_0, a_1, a_2$ in a power series for an initial value problem?

Assume $y=\sum a_n x^n$. The ODE is $$y'' + (2 - 4x^2)y = 0$$ $y(0) = 1, y'(0) = 0$ $a_0 = 1, a_1 = 0, a_2 = -1$
0
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1answer
300 views

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

Consider the following 2D infinitely large grid where the dots represent infinity: ...
13
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1answer
743 views

Does a power series vanish on the circle of convergence imply that the power series equals to zero?

Let $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be a power series, $a_n, z\in \mathbb{C}$. Suppose the radius of convergence of $f$ is $1$, and $f$ is convergent at every point of the unit circle. Question:If ...
8
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4answers
2k views

What's the idea behind the Taylor series?

I understand that they are viewed as approximations, but was that Taylor's original hope? Assuming that a function can be written as a power series seems to me to be a wild assumption, without some ...
12
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2answers
396 views

Finding the derivative of $x\uparrow\uparrow n$

I am trying to find a general derivative for the function: $f(x)=x^{x^{x^{...^{x}}}}$however to do that I must find $f^{\prime }$ and $f^{\prime \prime}$...etc. I am now trying to write down a general ...
9
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2answers
425 views

Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$

The following is a historical question, but first some background: Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be ...
7
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2answers
262 views

Prove $(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j} $

I stumbled upon the identity $$(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j}. $$ The right-hand side is a polynomial. ...
5
votes
2answers
280 views

Basic guidance to write a mathematical article.

I'm trying to put together a mathematical article on how to obtain certain infinite series for some well known functions by a method of integrals (I like to call it "The Integral Method" - thank you), ...
5
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1answer
129 views

what is the summation of such a finite sequence?

The summation is: $$\sum_{i=0}^n \binom{2i}i \binom{2n-2i}{n-i}$$ The answer is $4^n$. How to prove it, and how to think out it?
3
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2answers
247 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 $$ F(x)=\sum_{n=1}^{\infty}(-1)^n\Big(\sum_{m=0}^{\infty}(x^n)^m\Big)=\sum_{m=0}^{\infty}\Big(1-\frac{1}...
6
votes
2answers
550 views

Equality with Euler–Mascheroni constant

While trying to prove integral with exponential function and logarithm in an alternative way, I came to this solution: $$\sum_{k=0}^{+\infty}(-1)^{k+1}\frac{\log (k+1)+\gamma }{(k+1)}.$$ As both ...
2
votes
1answer
188 views

Summation of exponential series [duplicate]

Evaluate the limit: $$ \lim_{n \to \infty}e^{-n}\sum_{k = 0}^n \frac{n^k}{k!} $$ It is not as easy as it seems and the answer is definitely not 1. Please help in solving it.
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2answers
482 views

Finding the convergence interval of $\sum\limits_{n=0}^{\infty} \frac{n!x^n}{n^n}$.

I want to find the convergence interval of the infinite series $\sum\limits_{n=0}^{\infty} \dfrac{n!x^n}{n^n}$. I will use the ratio test: if I call $u_n = \dfrac{n!x^n}{n^n}$, the ratio test says ...
10
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1answer
321 views

Is this generalization of an exercise in Stein true?

The following question is exercise $14$ in chapter $2$ in Stein and Shakarchi's Complex Analysis. Suppose that $f$ is holomorphic in an open set containing the closed unit disc, except for a pole ...
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4answers
401 views

Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field

I would like to show that if $R$ is a field, then $R(x)$ is a proper subset of $R((x))$, where $R(x)$ is the ring of rational functions, and $R((x))$ is the ring of formal Laurent series. If $f \in ...
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votes
3answers
1k views

Puiseux series over an algebraically closed field

Using the construction $R_n = K[t^\frac1n]$, $L_n = \text{Quot}(R_n)$ and $P = \bigcup_{n\in \mathbb{N}}L_N$ one automatically gets that the Puiseux series are a field. Nevertheless they are also an ...
8
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2answers
1k views

Sum of sum of $k$th power of first $n$ natural numbers.

I was working on a problem which involves computation of $k$-th power of first $n$ natural numbers. Say $f(n) = 1^k+2^k+3^k+\cdots+n^k$ we can compute $f(n)$ by using Faulhaber's Triangle also by ...
5
votes
2answers
109 views

Short form of few series

Is there a short form for summation of following series? $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ $$\sum\limits_{n=0}^\...
5
votes
2answers
901 views

Singularities of $e^{z - \frac{1}{z}}$

I believe $e^{z - \frac{1}{z}}$ has essential singularities at $z = 0$ and $z = \infty$ (in both cases because of a $\frac{1}{z}$ in the exponent) but I'm having a hard time proving this. How can one ...
4
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1answer
294 views

closed form for a series over the Riemann zeta zeros

given the series $ \sum_{\rho} \frac{1}{z-\rho} $ here the sum is taken OVER the roots of the Riemann function on the critical line $ 0 < Re(s) <1 $ the summation is understood as we sum the pair ...
4
votes
5answers
240 views

Calculation of limit without stirling approximation

$\lim n^n/(e^nn!)=0$ using Stirling approximation it is obvious. But can we do it without using Stirling approximation. Now series with terms $x^n n^n/n!$ has ROC $1/e$. What we can say about ...
3
votes
2answers
210 views

Power series expression for $\exp(-\Delta)$

I know it should be true, but for some reason I can't get the calculations to work out in order to show that if $f$ is smooth and compactly supported, the power series $\sum_{j=0}^\infty \frac{(\...
2
votes
0answers
290 views

Prove that sum is finite with the help of generating function

Please help me to prove that the following sum is finite $$ \sum_{j=2l-2}^{\infty}j!\, a_j^{(l)}, $$ here the generating function of $\displaystyle{a_j^{(l)}}$ is $\displaystyle{\left(\frac{e^{x/6}\...
9
votes
1answer
105 views

What is $f_\alpha(x) = \sum\limits_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n$?

I want to understand the function $$f_\alpha(x) = \sum_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n, \ \ \ \forall x\in\mathbb{R},$$ for any possible real $\alpha\geq0$. I know that for $\alpha$ integer, $...
8
votes
4answers
428 views

Why do we say “radius” of convergence?

In an intuitive sense, I have never understood why a power series centered on $c$ cannot converge for some interval like $(c-3,c+2]$. Also, I have had a few professors casually mention that a series ...
8
votes
2answers
854 views

Power series without analytic continuation

Given a formal power series $\sum a_n z^n$ and a radius of convergence $R>0$, there are various ways to extend the function to the boundary such as Abel's theorem Fatou's lemma $H^\infty$ ...
7
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2answers
1k views

Continued fraction expansion related to exponential generating function

A recent SciComp.SE Question motivates us to ask for a nice continued fraction expansion of the following Maclaurin series: $$ f(x) = \sum_{n=0}^\infty \frac{B_n\; x^{n+3}}{n! (n+3)} = \int_0^x \frac{...
7
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3answers
2k views

Generalised Binomial Theorem Intuition

It was not until recently (why don't they teach it in secondary school?) that I've come across the Generalised Binomial Theorem, which from what I can tell is basically the same as the regular ...
6
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3answers
160 views

Is this action of $\mathbb F[[x]]$ on $\bigoplus_{i=0}^{\infty}\mathbb F$ natural?

The title of my question has a field $\mathbb F$ in it, but to make sure I'm not losing anything, I would like to introduce my question in full generality. But still, I will be happy with an answer in ...
5
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3answers
591 views

Compositions of $n$ with largest part at most $m$

This is a problem from Stanley's Enumerative Combinatorics that I'm failing at a bit (lot): Let $\bar{c}(m,n)$ denote the number of compositions of $n$ with largest part at most $m$. Show that $$\...
4
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1answer
158 views

Estimate the scale of $e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m}$

I would like to estimate the scale of the following series, $$S(m,t)=e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m},$$ where $e$ is the base of ...
4
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1answer
378 views

Formal Power Series — what's in it?

I have the following statement in a paper: Let $\Psi$ be the formal power series defined over the alphabet $\Omega$ and the log semiring by: $(\Psi, (a, b)) = -log(c((a,b)))$ for $(a,b) \in \Omega$, ...
4
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2answers
549 views

Graphical interpretation of infinite power series?

Can someone please give me a graphical interpretation/sense of infinite power series? Some functions such as exponentials, sines, and cosines are infinite power series, but what does that mean and ...
3
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2answers
6k views

Finding the power series of $\arcsin x$

I'm trying to find the power series of $\arcsin x$. This is what I did so far: $(\arcsin x)'=\frac{1}{\sqrt{1-x^2}}$, so $\arcsin x=\int \sqrt{\sum_{n=0}^{\infty}x^{2n}}$. (for $|x|<1$) Any ...
3
votes
3answers
3k views

Power (Laurent) Series of $\coth(x)$

I need some help to prove that the power series of $\coth x$ is: $$\frac{1}{x} + \frac{x}{3} - \frac{x^3}{45} + O(x^5) \ \ \ \ \ $$ I don't know how to derive this, should I divide the expansion of $...
3
votes
1answer
783 views

Power Series With Bernoulli Numbers

The exercise reads "Express the power series for $\large \frac{z}{\sin (z)} = \frac{2 i z}{e^{iz} - e^{-iz}} $ in terms of Bernoulli numbers." I am given in a previous exercise that the Bernoulli ...
2
votes
3answers
524 views

Simple Power Series Expansion for Problems similar to $f = (1 + \epsilon \,x)^{1/\epsilon}$

I was flicking through a book on perturbation methods and saw a simple question asking the reader to expand the following expression for $f$ in a power series (up to the first 2 terms): $f = (1 + \...
2
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4answers
154 views

Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $

Evaluate: $$\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $$ I rewrote the sum as $$\sum_{n=0}^{\infty} \frac {1}{4^{8n-7}(8n-7)} - \sum_{n=0}^{\infty} \frac {1}{4^{8n-3}(8n-3)}$$ Now, I ...
2
votes
2answers
157 views

Understanding Power Series Multiplication Step

Working on Spivak's Calculus problems, I searched online, trying to understand the solution provided for Problem 4a of Chapter 2. I found the question I needed: Spivak's Calculus - Exercise 4.a of ...
12
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6answers
3k views

Why infinity multiplied by zero was considered zero here?!

I watched an online video lecture by some professor and she was solving a convergence problem of the power series $$\sum_{n=1}^\infty n!x^n,$$ i.e., she was finding the values of $x$ for which this ...
9
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3answers
498 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 ...
9
votes
1answer
353 views

Does a convergent power series on a closed disk always converge uniformly?

If I have a power series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i} \in\mathbb{C}[[z]]$ with radius of convergence $r>0$ and I know that the series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
5
votes
1answer
1k views

How does a complex power series behave on the boundary of the disc of convergence?

Until recently, I thought my math teacher had said in his real analysis class that on the boundary of the disc of convergence, a complex power series has at least one point at which it does not ...
5
votes
1answer
271 views

The net signed area between $t=0, y=0, t=x$, and $y=f(t)$

f(t) is continuous function.So I know that $\int _0^x {f(t) dt}=$ "The net signed area between $t=0, y=0, t=x$, and $y=f(t)$" And I can find the same result with endless small rectangulars areas ...
5
votes
1answer
443 views

When is a Fourier series analytic?

By Fourier theory, every continuously differentiable function $f : S^1 \to \mathbf C$ admits a unique, uniformly convergent Fourier expansion $$f(\theta) = \sum_{n\in \mathbf Z} a_n e^{in\theta}.$$ ...