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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2
votes
1answer
133 views

Choice of the First Term in Legendre Polynomials

The two solutions of the Legendre's Differential Equation obtained by series solution method are : and Now according to my textbook, for the useful polynomial for n equal to a positive integer, ...
2
votes
1answer
10k views

Maclaurin Series for $\arctan(x)$ by successive differentiation

I am trying to find a Maclaurin Series for $\arctan(x)$ up to the term with the fifth power of x and I have to use the method of successive differentiation. I know (from an example in my notes) the ...
1
vote
1answer
39 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 ...
0
votes
3answers
54 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?
0
votes
2answers
87 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 ...
9
votes
2answers
414 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 ...
8
votes
4answers
1k 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 ...
5
votes
2answers
276 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), ...
12
votes
2answers
384 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 ...
7
votes
2answers
252 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
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
votes
2answers
241 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 $$ ...
16
votes
2answers
465 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
votes
1answer
304 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 ...
4
votes
1answer
286 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 ...
2
votes
1answer
156 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.
8
votes
2answers
712 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
votes
2answers
961 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 ...
6
votes
2answers
411 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
2answers
108 views

Short form of few series

Is there a short form for summation of following series? $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ ...
5
votes
3answers
559 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 ...
5
votes
2answers
850 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 ...
5
votes
2answers
527 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 ...
4
votes
5answers
236 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
206 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 ...
3
votes
1answer
712 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 ...
3
votes
2answers
5k 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 ...
2
votes
4answers
145 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
0answers
289 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 ...
10
votes
1answer
372 views

Can the result of termwise multiplication of power series be found in a closed form?

This is a question that comes out of a combinatorics question that is using generating functions. Let me define what I mean by multiplying power series the wrong way (although you may be able to ...
9
votes
1answer
103 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
376 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 ...
7
votes
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
votes
2answers
866 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 ...
6
votes
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
votes
2answers
181 views

What is the order of this pole?

$$f(z)=\frac 1{\cos(z^4)-1}$$ $z=0$ is a pole of $f$, and I believe that the Laurent series centred at $0$ is $-\frac 2{z^8}-\frac 16+...$, which looks like the pole is of order $8$, but why does ...
5
votes
2answers
333 views

Where to find reference about dealing with operators in form of formal power series?

I often encounter the following statements: $${D \over e^D - 1} = {\log(\Delta + 1) \over \Delta}$$ $$\int_x^{x+1} f(t)\,dt= {e^D - 1 \over D} [f]$$ $$\Delta = (e^D - 1)\,$$ $$f(a+x)=e^{a D}[f]$$ ...
5
votes
1answer
270 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
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
3answers
430 views

A deceiving Taylor series

When we try to expand $$ \begin{align} f:&\mathbb R \to \mathbb R\\ &x \mapsto \begin{cases} \mathrm e^{-\large\frac 1{x^2}} &\Leftarrow x\neq 0\\ 0 &\Leftarrow x=0 ...
4
votes
2answers
440 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 ...
4
votes
1answer
337 views

How to properly translate the coefficients of a Taylor series?

Given a Taylor series $$f(z) = \sum_{k=0}^\infty c_k^{(a)}\frac{(z-a)^k}{k!}$$ of a meromorphic function $f$ in $\mathbb C$ (i.e. analytical except for a set of isolated points) around some value ...
4
votes
1answer
157 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
votes
1answer
366 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 $(a,b) \in \Omega$, ...
3
votes
4answers
123 views

First four terms of the power series of $f(z) = \frac{z}{e^z-1}$?

Attempt: $$ e^z = \sum_{n=0}^\infty \frac{z^n}{n!}$$ $$ e^z - 1 = \sum_{n=0}^\infty \frac{z^n}{n!} -1$$ $$ e^z - 1 = z\sum_{n=0}^\infty \frac{z^n}{(n+1)!} $$ Thus $$ \frac{z}{e^z-1} = ...
3
votes
2answers
242 views

If it converges, how to show that power series converges to $f(x)$?

I had a very basic question. Suppose $f(x)$ is a function. And let us say it has a power series :- $$f(x) = \sum_{n=0}^\infty a_nx^n.$$ Suppose we are operating inside the region of convergence. ...
3
votes
3answers
2k 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
5answers
425 views

Calculate sum of an infinite series

I have been struggling with this functional series. $$\sum_{n=1}^{\infty}{(-1)^{n-1}n^2x^n}$$ I need to calulate the sum.Any tips would be appreciated.
2
votes
0answers
221 views

Algorithm for reversion of power series?

Given a function $f(x)$ of the form: $$f(x) = x/(a_0x^0+a_1x^1+a_2x^2+a_3x^3+a_4x^4+a_5x^5+...a_nx^n)$$ Let $A$ be an arbitrary (any) infinite lower triangular matrix with ones in the diagonal: $$A ...
2
votes
2answers
134 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 ...