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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3
votes
1answer
93 views

Is $\frac{1}{2^{2^{0}}}+\frac{1}{2^{2^{1}}}+\frac{1}{2^{2^{2}}}+\frac{1}{2^{2^{3}}}+…$ algebraic or transcendental?

Inspired by this question, the series $\dfrac{1}{2^{2^{0}}}+\dfrac{1}{2^{2^{1}}}+\dfrac{1}{2^{2^{2}}}+\dfrac{1}{2^{2^{3}}}+\dots$ is clearly irrational. But is it algebraic or transcendental? I ...
1
vote
0answers
25 views

Problem with custom made natural log and power functions

I have made these two functions with the help of posts on math.stackexchange.com. For ln I'm using information gathered from Calculate Logarithms by Hand and for ...
0
votes
2answers
31 views

Solving ODEs via power series - what is wrong with my solution?

I need to solve the ODE $x^2y''+xy'+(x^2-\frac 1 4 )y=0$. The solution I should get is $y(x)=x^{\frac{-1}2}\sin x$, but using power series, I got all the coefficients are zero. Here is my solution- ...
5
votes
2answers
102 views

How do i evaluate this sum :$\sum _{m=1}^{\infty } \sum _{k=1}^{\infty } \frac{m(-1)^m(-1)^k\log(m+k)}{(m+k)^3}$?

How do I evaluate the following sum: $$\sum _{m=1}^{\infty } \sum _{k=1}^{\infty } \frac{m(-1)^m(-1)^k\log(m+k)}{(m+k)^3}$$ Note I used many idea such as :Hochino's Idea and taylor expansion of ...
0
votes
0answers
16 views

Uniform and ordinary convergence of a series.

I have a series of the kind $\sum\limits_{n=1}^{\infty} \frac{a_nt^n}{1+t^{2n}}$, $t\in(0,\infty)$. Btw, $a_n$ are real and they are determined by several integrals, but it is possible to compute any ...
4
votes
1answer
175 views

Prove the identity $\cosh(2x)=\cosh^2(x)+\sinh^2(x)$ using the Cauchy product. [closed]

Prove the identity $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ using the Cauchy product and the Taylor series expansions of $\cosh(x)$ and $\sinh(x)$. The relations involving the exponential function are ...
1
vote
0answers
36 views

Relations between convergence on boundary of power series, and the uniformity of convergence

Given a power series $\sum_{n=0}^\infty a_n x^n \ ,x \in \mathbb R\ $ with radius of convergence $R$. Is that true that: If the series does not converge at one boundary, $R$ or $-R$, then the ...
3
votes
2answers
100 views

Ordinary generating function of powers of 2

Is there a good closed form expression for the generating function of the formal power series $$ A(z) := \sum_{n=0}^\infty z^{2^n} = z + z^2 + z^4 + z^8 + z^{16} + \cdots. $$ Is there a tractable way ...
0
votes
0answers
30 views

Express $\cos^2\theta\cos\phi\sin\phi$ in Spherical Harmonics

I am looking for a form of $$\cos^2\theta\cos\phi\sin\phi=\sum_{lm}c_{lm}Y_l^{m}(\theta,\phi),$$ where $Y_{lm}$ is the spherical harmonics. The idea I believe would be to find ...
1
vote
2answers
27 views

Taylor series for multivariable functions

To expend the function of multiple variables $$ f({\bf x})=f(x_1,x_2,\dots,x_n):\mathbb R^n\to\mathbb R $$ in Taylor series around $\bf 0$, we have $$ f({\bf x})=f({\bf 0})+Df({\bf 0})\cdot{\bf ...
1
vote
1answer
41 views

How to expand the Taylor series of functions of several vectors?

We know that the Taylor series expansion of the function of several scalars around zero is $$ f(x,y)=f(0,0)+f_x(0,0)\cdot x+f_y(0,0)\cdot y+\frac{1}{2!}f_{xx}(0,0)\cdot x^2+\dots $$ Then, how about ...
0
votes
2answers
43 views

Find the series expansion of $\text{csch}^{-1}(x)$

Find the series expansion of $\text{csch}^{-1}(x)$ $\text{csch}^{-1}(x)=\ln 2-\ln x+\frac{x^2}{4}-\frac{3}{32}x^4+\frac{5}{96}x^6-...$ $\text{csch}^{-1}(x)=\ln(\sqrt{1+\frac{1}{x^2}}+\frac{1}{x})$ ...
3
votes
1answer
52 views

Convergence of a series in $R^2$

For $(x,y)\in\mathbb R^2$, consider the series $$ \lim_{n→\infty}\sum_{l,k=0}^n\frac{k^2x^ky^l}{l!}. $$ Then the series converges for $(x,y)$ in $(-1,1)\times(0,\infty)$ $\mathbb R\times(-1,1)$ ...
1
vote
2answers
28 views

What is the radius of convergence of $\sum\limits_{k=0}^\infty \sin k\,x^k$?

This is just a power series I came up with. I have plotted the series for up to $100$ terms and the plot seems to be pretty stable between over $(-1,1)$, but I'd like to verify that with a convergence ...
1
vote
3answers
76 views

Series Solution to $y''+xy=e^x$

I am thoroughly familiar with using power series to solve the differential equation $y''+xy=0$, but how exactly does one go about solving $y''+xy=e^x$? I would imagine you represent $e^x$ as it's ...
1
vote
0answers
43 views

Convergence of some sums of complex functions

In the context of the probability theory of rare events i found myself dealing with these series of complex functions: $\sum_{n=1}^\infty(1+n)^{-k}z^{n^2}\\$ with z Complex and k Real. ...
1
vote
0answers
22 views

Irrationally termed converging infinite series

Some power series consist of an infinite number of rational terms converging to an irrational limit. Is there a series expansion /expression built on terms of powers of $\pi$ or $e$ summing up to 1 ...
0
votes
0answers
33 views

Different forms of remainder in Taylor series

In the literature, it is common to find different forms of the remainder function in a truncated Taylor series. To name a few: Integral form Lagrange form Cauchy form First, can you tell me any ...
1
vote
2answers
32 views

Power series with square summable coefficients

An exercise in my textbook asks to prove the following: Suppose $\sum |a_n|^2<\infty$. Show that the power series $\sum a_nz^n$ has radius of convergence at least equal to $1$. My reasoning ...
1
vote
0answers
104 views

Can be justified $\zeta(3)=\lim_{n\to\infty}-3\sum_{k=1}^n\sum_{\nu=0}^\infty\frac{(-1)^{\nu+1}}{\nu+1}\binom{3k^3-1}{\nu}$?

My main goal is understand useful facts about my computations, that could be wrong, the way shold be too a street without exit looking for a evaluation of Apéry's constant, since I don't use any ...
1
vote
2answers
58 views

Prove that if $\lim_{n\rightarrow \infty} |\frac{c_n}{c_{n+1}}|$ exists, then it is equal to the radius of convergence

a) Prove that if $\lim_{n\rightarrow \infty} | \frac{c_n}{c_{n+1}}|$ exists, then it is equal to the radius of convergence of $\sum_{n=0}^{\infty}c_n x^n$. First assume that $\lim_{n\rightarrow ...
2
votes
1answer
43 views

A Power Series Solution to a differential equation.

Find a power series solution to the following equation: $$z''=\frac{4}{t}z'+z=0$$ which is bounded by $t=0$ with $z(0)=1$, $z'(0)=0$. So far this is what I have: $z=\sum_{n=0}{c_nt^n}$ ...
2
votes
1answer
51 views

Real analytic way to explain why the radius of convergence of $1/(1+x^2)$ is small

For any series expansion of $\frac{1}{1+x^2}$, the disc of convergence is blocked by the two singularities on $+i$ and $-i$. A series expansion about $0$ gives a radius of convergence of $1$. Is ...
0
votes
1answer
31 views

Prove that there is a positive number $\delta$ such that the sum of the series is nonzero

Let $a, c_0, c_1,\ldots \in\mathbb{R}$ with at least one of $c_0, c_1,\ldots$ nonzero. Let $\sum_{n=0}^{\infty}c_n (x-a)^n$ be a power series with radius of covnergence $r>0$. Prove that ...
0
votes
4answers
39 views

Showing a power series converges…

I'd like to show that $$\sum\limits_{n = 1}^\infty {{{{x^{n + 1}}} \over {n(n + 1)}}} $$ absolutely converges for $|x| < 1$
0
votes
1answer
40 views

Use geometric progression formula to expand generating function into a power series?

I am a software engineer and I am studying combinatorics on my own to enhance my learning. I have been finally starting to get the hang of generating functions, but the following problem below has me ...
10
votes
1answer
142 views

Evaluation of $\sum_{n=0}^{\infty}\frac{1}{(n^4+n^2+1)n!}$

I am wondering how to evaluate the following sum: $$\sum_{n=0}^{\infty}\frac{1}{(n^4+n^2+1)n!}.$$ In wolfram alpha I find it is equal to $e/2$ . I have used the residue method but I didn't succeed ...
0
votes
1answer
25 views

Frobenius method confusion?

The question is: We are supposed to find the first 4 terms in the power series. I was able to work through the question but the my answer is wrong, the correct answer is supposed to be the ...
-1
votes
1answer
32 views

Limit of expression with increasing exponent

I got stuck trying to evaluate: $$\lim\limits_{n\to \infty}\left(\frac{\sqrt2}{\sqrt2+1}\right) + \left(\frac{\sqrt2}{\sqrt2+1}\right)^2 + \left(\frac{\sqrt2}{\sqrt2+1}\right)^3 + \dots + ...
1
vote
1answer
27 views

Finding the geometric series of the fraction

I am confused as to how to turn a fraction into a sum using geometric series. I have $\frac{z+2}{(z-1)(z-4)}=\frac{2}{z-4}+\frac{-1}{z-1}$ I do not know how I turn the last 2 fractions into ...
0
votes
2answers
28 views

Laurent series expansion of a complex function: $\frac{(z+1)}{z(z-4)^3}$

Find the Laurent series for $\frac{(z+1)}{z(z-4)^3} \in 0 < |z-4| < 4$. I get you have to write the denominator in another way, but what are the intermediate steps?
0
votes
0answers
14 views

Continuous functions that are the sum of a power series on a ball

Let $D$ be the closed unitary ball of $\Bbb C$ and $\Gamma$ the circle of radius $1$. Let $E$ be the vector space of the maps $f:D\longrightarrow \Bbb C$ which are both continuous and the sum of a ...
1
vote
1answer
41 views

Proof of rule of exponents

I'm trying to prove that $e^{-x}=1/(e^{x})$ using power series. Here's what I have thus far. I want to show that $\sum_{n=0}^{\infty} \frac{(-x)^n}{n!} = (\sum_{n=0}^{\infty} ...
0
votes
1answer
46 views

Why may we suppose $a=0$ in this proof?

I didn't understand why we may suppose $a=0$ in this proof: I'm reading Conway's Complex Analysis book, page 31. Any help is welcome
1
vote
1answer
63 views

Write $\frac {1}{1+z^2}$ as a power series centered at $z_0=1$

I'm trying to solve a question where I need to write $\frac {1}{1+z^2} $ as a power series centered at $z_0=1$ I'm not allowed to use taylor expansion. So my first thought was to rewrite the function ...
0
votes
1answer
21 views

Smaller radius of convergence

Suppose we have two power series, with coefficients $a_n, b_n$, both with radius of convergence, $R$. Let the power series with the coefficients $a_n+b_n$, with radius $R'$ I need an example for ...
1
vote
1answer
115 views

Generalization of Maclaurin series?

The Maclaurin series for a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{f^{(n)}(0)x^n}{n!}$$ Suppose that instead of the $x^n$ we picked up a function $g_n$? We can write ...
1
vote
4answers
88 views

Closed form for $\sum_{n=0}^{\infty} \binom{n+k}{k} x^n$

I was wondering if there is also a closed expression for the series $$\sum_{n=0}^{\infty} \binom{n+k}{k} x^n$$ where $|x|<1.$ A few examples suggest that the answer is $\frac{1}{(1-x)^{k+1}}$ ...
2
votes
1answer
40 views

Computing the Taylor expansion of the square root of cos(z),

Let $\large f(z)=\sqrt{cosz}$ with the branch of the square root chosen so that $f(0)=1$. Consider the power series expansion of $f(z)$ in powers of $z$. Part 1) Compute the first three non-zero ...
2
votes
1answer
51 views

Is $e^{x}$ algebraic over $\mathbb{C}(x)$?

We have $$e^{x}= \sum \frac{x^{n}}{n!}$$ as a power series expansion. We can see that such a power series has radius of convergence infinity and as a result $e^x$ is defined over the whole complex ...
2
votes
2answers
39 views

Elements of $\mathbb{C}(x)$ and algebraic elements over $\mathbb{C}(x)$

Well, the elements of the ring $\mathbb{C}[x]$ are easy to understand for me. They can be thought of as polynomial functions from $\mathbb{C} \rightarrow \mathbb{C}$ and as a result they are ...
1
vote
1answer
36 views

Doubts related to Differential Operator of Infinite order

Let $$f(s)=\sum_0^\infty c_vs^v$$ be some entire function. We say that the differential operator $f(d/dx)=\sum_0^\infty c_vd^v/dx^v$ is defined in some fundamental space $\varPhi$, if for any $\varphi ...
0
votes
1answer
24 views

How to show that $1+ \sum \limits _{n=1} ^\infty \frac {x^n} n$ converges pointwise?

I am having trouble showing that the taylor series for $-\ln(1-x)$ converges pointwise on $[0,1)$. I have that the $k$ derivative is $\dfrac {(k-1)!} {(1-x)^k}$. This gives that the Taylor series ...
0
votes
0answers
17 views

continuity of Power Series on Boundary

Let $\sum_0^\infty$ $a_k$ be a real convergent series. Show that the power series $f(x) := \sum_0^\infty a_k x_k$ is continuous on $[0, 1]$. I know that within the radius of convergence it is true by ...
0
votes
1answer
31 views

Power series and differentiation

I have to prove that this series $$ p(x)=\sum_{k=1}^\infty kx^k $$ converges for $x=(\frac67)$ and then find the value of $p(\frac67)$. For the first one I used the formula to find the radius of ...
4
votes
1answer
56 views

I am only able to show this upper bound for $|x|<1$. But how to show it for all $x$?

EDIT: I think that I have now solved it, but I have three constants $C_{\alpha}$, one for the case |x|<1 (the easiest case), one constant for the case $|x|\ge 1$ but summing the series only from 0 ...
0
votes
1answer
19 views

Find the power series for the function

How do I find the power series for f(x)=$\frac{x^3}{x+3}$ ? That to in summation notation.
2
votes
2answers
91 views

Definite integral of $e^{x/2}$ using Maclaurin polynomial

My professor asked us to find the 3rd degree Maclaurin polynomial of $e^{x/2}$ which I found to be $$1 + \frac{x}{2} + \frac{x^2}{8} + \frac{x^3}{48}$$ I do know that that the series for ...
1
vote
1answer
43 views

Prove the series converges to a continuous function

Consider the series $\sum_{k=1}^\infty \frac{1}{x^2+k^2}$ Show that this series converges to a continuous function that is defined for all $x ∈ R$. I'm unsure how to approach this. I'm ...
0
votes
2answers
54 views

Find Taylor Series of $\frac{1}{1+z^2}$ around $1$

For $f(z)=\dfrac{1}{1+z^2}$ find the Taylor series centered at $1$. While I know I could use partial fractions or perhaps maneuver this problem by adding constants, I would really like to use the ...