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
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2answers
147 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 ...
2
votes
3answers
518 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 + ...
1
vote
3answers
103 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??
12
votes
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 ...
10
votes
1answer
390 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
342 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
404 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 $$f(\theta) = \sum_{n\in \mathbf Z} a_n e^{in\theta}.$$ ...
5
votes
2answers
184 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
342 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
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
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
443 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
3answers
156 views

the sum of $1-\frac{1}{5}+\frac{1}{9}-\frac{1}{13}+…$

I thought this was the real part of the series: $\sum_{n=0}^\infty \frac{i^n}{1+2n}$, with $i=\sqrt{-1}$. When taking the real part I am left with: $\sum_{n=0}^\infty \frac{\cos(n\pi/2)}{1+2n}$. I ...
4
votes
1answer
359 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
2answers
150 views

What's the background of this exercise?

I found this interesting exercise on a calculus book (Stewart) Let $$ u=1+\frac{x^3}{3!}+\frac{x^6}{6!}+\cdots $$ $$ v=x+\frac{x^4}{4!}+\frac{x^7}{7!}+\cdots $$ $$ ...
4
votes
1answer
3k views

Finding closed forms for $\sum n z^{n}$ and $\sum n^{2} z^{n}$

Using the identity $\frac{1}{1-z} = 1 + z + z^2 + \ldots$ for $|z| < 1$, find closed forms for the sums $\sum n z^n$ and $\sum n^2 z^n$. My solution: Because $\displaystyle1 + z + z^{2} + ...
4
votes
2answers
252 views

Proving the equivalence of a sum and a double integral

Based on "Certain Subclass of Starlike Functions" journal by Chun-Yi and Shi-Qiong Zhou in 2007 (Science Direct), I found difficulties to understand the proof in Theorem 3 where they have verified: ...
4
votes
3answers
258 views

Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$

Given the geometric series: $1 + x^2 + x^4 + x^6 + x^8 + \cdots$ We can recast it as: $S = 1 + x^2 \, (1 + x^2 + x^4 + x^6 + x^8 + \cdots)$, where $S = 1 + x^2 + x^4 + x^6 + x^8 + \cdots$. This ...
3
votes
1answer
39 views

Radii of convergence for complex series

I need to find the radii of convergence for these series: $1. \sum_{n=1}^\infty (2+(-1)^n)^n z^{2n}$ $2. \sum_{n=1}^\infty (n+a^n)z^n, a \in C $ $3. \sum_{n=1}^\infty 2^n z^{n!}$ Starting with ...
3
votes
4answers
141 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
1answer
1k views

Is there a simple proof that a power series can be differentiated term by term?

I know that the derivative of the sum of a power series can be calculated by summing the derivatives of the terms, and that the resulting series has the same radius of convergence as the original. The ...
3
votes
1answer
548 views

Show that some $C^\infty$ real function is analytic

Consider the function $f:(a,b) \rightarrow \Bbb R $, which is $C^\infty$ class. It is required to show that if the function $f$ satisfies: $\forall_{n\in\Bbb N_+} \forall_{x\in(a,b)} f^{(n)}(x) \ge 0 ...
3
votes
2answers
818 views

How to bound the truncation error for a Taylor polynomial approximation of $\tan(x)$

I am playing with Taylor series! I want to go beyond the basic text book examples ($\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, etc.) and try something different to improve my understanding. So I ...
3
votes
2answers
753 views

How to compute coefficients in Trinomial triangle at specific position?

I need to compute coefficients of $i$-th power of $x$ from simplifying of $$(x^2 + x + 1)^n$$ From that site i know about trinomial triangle. But how can compute coeficients of $i$-th element at ...
3
votes
5answers
428 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
2answers
45 views

How to do power series expansion

What is the coefficient of $x^{11}$ in the power series expansion of $\frac 1{1-x-x^4}$? How do I do power series expansions?
2
votes
1answer
40 views

limit of $a_n(x)=\frac{\sum_{k=0}^{t_n}x^k}{\sum_{k=n+1}^{t_n}x^k}$

Let $t>1$ and $[t_n]$ defines the integer part of the product $t.n$. I need to evaluate the sequence $$a_n(x)=\frac{\sum_{k=0}^{[t_n]}x^k}{\sum_{k=n+1}^{[t_n]}x^k}$$ is convergent or bounded as ...
2
votes
2answers
260 views

Evaluation morphisms of formal power series and nilpotent elements

Given a commutative ring $A$, and a finitely presented (associative) $A$-algebra $B$, show that a morphism of $A$-algebras $A[[x]] \longrightarrow B$ is given by evaluation at an nilpotent element $ ...
2
votes
0answers
240 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
1answer
103 views

Power series method to solve Airy’s differential equation [duplicate]

Using power series method, solve Airy’s equation $$y′′+ xy = 0.$$ How do I start solving this? Thanks in advance!
2
votes
2answers
1k views

Using Generating Functions to Solve Recursions

I have the recursion $A(n) = A(n-1) + n^2 - n$ with initial conditions $A(0) = 1$. I attempted to solve it using generating functions and I'm not quite sure I have it right, so I thought I might ask ...
2
votes
1answer
5k views

How to find the general solution of $(1+x^2)y''+2xy'-2y=0$. How to express by means of elementary functions?

Find the general solution of $$(1+x^2)y''+2xy'-2y=0$$ in terms of power series in $x$. Can you express this solution by means of elementary functions? I know that $y= ...
2
votes
2answers
596 views

Help on differential equation $y''-2\sin y'+3y=\cos x$

$y''-2\sin y'+3y=\cos x$ I'm trying to solve it by power series, but I just can't find the way to get $\sin y'$. Is there any special way to find it?
2
votes
2answers
1k views

${a_n}$ series of Fibonacci numbers. $f(x)=\sum_{0}^{\infty}a_nx^n$, show that in the convergence radius: $f(x)= \frac{1}{1-x-x^2}$

I'd really like your help with this following problem: Let ${a_n}$ be a Fibonacci series $a_1=a_0=1$ and $a_{n+2}=a_n+a_{n+1}$ for every $n \geq 0$. Let $f(x)=\sum_{0}^{\infty}a_nx^n$, I need to find ...
1
vote
1answer
31 views

Translations AND dilations of infinite series

Sometimes, when working with infinite series, it's useful to add "dilated" or "translated" versions of the infinite series, term by term, back to the original. There are ways of making this rigorous ...
1
vote
1answer
88 views

limit of jacobi theta 2 or simple series

I have a simple problem: I need to evaluate the limit $x\rightarrow 1$ of the Jacobi Theta function 2 $$\Theta_2(m,x)=2x^{1/4}\sum_{k=0}^\infty x^{k(k+1)}\cos((2k+1)m)$$ when $m=0$, that to say ...
1
vote
2answers
46 views

Finding the co-efficients of this power series

I am required to find the co-efficients of this power series: $2x\ln(1+2x)$ I approached the problem by considering the $\ln(1+2x)$ part as the integral of $2/(1+2x)$ and applied the geometric series ...
1
vote
1answer
62 views

Derivative of exp with definition of differentiability

Prove with the definition of differentiability that $\exp(z)$ is differentiable in $\mathbb C$ and $(\exp(z))' = \exp(z)$ for all $z \in \mathbb C.$ I tried: \begin{align*} \frac{\exp(z+h) - ...
1
vote
0answers
68 views

Radius of convergence of $\sum_{n\ge 1}{\frac{x^{n^2}}{n^2}}$

I want to determine the radius of convergence of the power series $$\sum_{n\ge 1}{\frac{x^{n^2}}{n^2}}$$ Is my following try correct, and is there any simpler way to do this: Put ...
1
vote
1answer
258 views

Applications of higher powers of trigonometric functions

I am after a reference (book, papers etc) about the practical applications of trigonometric functions raised to higher powers. An example is one that I have been using in my own studies: $\cos^4 ...
1
vote
2answers
99 views

What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$

For a nonnegative integer $b$, and $|x|<1$, what is the function given by the power series $$ \sum_{n=0}^\infty \binom{b+2n}{b+n} x^n. $$ For $b=0$, this post shows $$ \sum_{n=0}^\infty ...
1
vote
1answer
3k views

Radius of convergence for the exponential function

I'm studying physics and am currently following a course on complex analysis and in the section on analytic functions, the radius of convergence $R$ for power series was introduced. The Taylor ...
1
vote
1answer
593 views

The coefficients in the inverse of unit lower triangular Toeplitz matrix

I have a question about the inverse of lower triangular Toeplitz matrix. There is a matrix $Q$ which can be infinite-dimensional. $$ Q=\left[\begin{array}{ccccc} 1\\ -k_{1} & 1\\ -k_{2} & ...
0
votes
2answers
44 views

When is a finite sum of powers of non-integer a rational number? [closed]

Concretely, is there $ b \in \mathbb R, n,k \in \mathbb N $ such that $ \sum_{i = n}^{n+k} b^i \in \mathbb Q$ ?
0
votes
2answers
51 views

Could the sum of powers of non-integers result in a whole number? [closed]

Concretely, is there a $ b \in \mathbb R $ such that $ \sum_{i \in I \subset \mathbb N} b^i \in \mathbb W$ ?
0
votes
1answer
58 views

How to solve $xy''+2y'+\lambda^2 \, xy=0$ with the power series method?

Find all functions $y(x)$ which can be expressed as a convergent power series $y(x)=\sum_{n=0}^{\infty} a_n x^n$ and which satisfy the following differential equation: $$xy''+2y'+\lambda^2 \, xy=0$$
0
votes
2answers
119 views

radius of convergence $\displaystyle \sum_{n=1}^{\infty}n! x^{n!}$

I just wondering radius of convergence following series $$ \sum_{n=1}^{\infty}n! x^{n!} \\ $$ My 1st attempt is 'root test' $$ \sqrt[n!]{|a_{n!} |} =\sqrt[n!]{|n! |} =\sqrt[t]{t} \rightarrow 1 $$ So, ...
0
votes
1answer
151 views

Find the radius of convergence of the power series

$\displaystyle\sum_{n=0}^{\infty}a_nz^n$, where $a_{2k+1} = 2^k$ and $a_{2k} = (1 + (1/k))^2$ for $k = 0, 1, 2, \dotsc$ I started off by doing the ratio test, but I know that the ration test is for ...
0
votes
2answers
621 views

Behavior of $\sum_{n=1}^\infty n^{-1}z^n$ on the circle of convergence

Consider the following complex power series :$$\sum_{n=1}^\infty\frac{z^n}{n}$$ The radius of convergence of this series is $1$ and the series is divergent for $z=1$. I want to know what are the ...
0
votes
1answer
176 views

What's the limit of coefficient ratio for a reciprocating power series?

I have a question about the coefficient in the inverse of the power series. Assume $$ f=1-\sum_{i=1}^{\infty}(ck_i)x^i, $$ where $c$ and $k_i$ are positive and $0<ck_i<1$ for any $i>0$. ...