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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0
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2answers
46 views

Show that the radius of convergence of $e^x$ is infinite

I am a bit confused as to whether I am doing this question correctly. Firstly, we have defined the radius of convergence of a power series centered at a $$\sum_{n=0}^{\infty} a_n(x-a)^n$$ to be the ...
1
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1answer
33 views

Division between power series that converge at least for |x| < r, only valid for |x| sufficiently small?

I have this book Calculus, Ninth Edition by Varberg, Purcell, and Rigdon; there's a particular point of a theorem (and another line after that) about Infinite Series that I really don't understand. I ...
1
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1answer
23 views

Evaluating Maclaurin Series

I would like to know how they got the highlighted part in the image below; What I have done so far is, finding the Maclaurin series for $e^x$ then substitute $2x$ for $x$ and find the Maclaurin series ...
0
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2answers
36 views

Convergence of Sum Involving Double Factorial

I have the sum $$\sum_{I=1}^\infty a^i (2i-1)!!$$ where $!!$ is the double factorial (the product of all the integers from 1 up to some non-negative integer n that have the same parity as n is called ...
0
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1answer
21 views

complexity of building heap: why can one substitute a bounded infinite series into a bounded sum?

Partially into the derivation, the author substitutes the result of this infinite series, $$ \sum_{h=0}^\infty hx^h = \frac{x}{(1-x)^2} $$ into the bounded sum, $$\sum_{j=0}^h ...
3
votes
1answer
58 views

How to prove that this complex series converges?

In Conway's book, "Functions of one complex variable", page 33 ex. 7., ask for the convergence of the series $$\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n}z^{n(n+1)},\ \ \ \text{when } z=i.$$ I know that if ...
1
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2answers
33 views

Find the solution of a differential equation in the form of a power series

Find the solution of the differential equation $y''(x)=y(x)$ with $y(0)=1$ and $y'(0)=0 $ in the form of the power series $y(x)=\sum_{j=0}^{\infty}a_jx^j$ Use the following method of ...
0
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2answers
27 views

Radius of convergence of $\frac{x}{sinh(x)}$

The power series representation of real hyperbolic sine function, as $sinh(x)= \sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}$. And its radius of convergence is, of course, $\frac{1}{\lim_{n\to \infty} ...
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0answers
39 views

OEIS A249665 generating function

I'm stuck at finding the general term of the sequence $$1, 1, 1, 2, 6, 14, 28, 56, 118, 254, 541, 1140, 2401, 5074, \ldots$$ According to OEIS, Colin Barker conjectured the recurrence relation to be ...
10
votes
1answer
373 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 ...
5
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1answer
220 views
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2answers
61 views

How to calculate sum of power series? [duplicate]

I'm trying to work out sum of this series $$1 + \frac{2}{2} + \frac{3}{2^2} + \frac{4}{2^3} + \ldots$$ I know one method is to do substitutions and getting the series into a form of a known series. ...
2
votes
2answers
105 views

How to determine $\sum_{k=0}^{\infty}\frac{1}{(5k+1)^{2}}$?

Is there way to determine this sum below? $$\sum_{k=0}^{\infty}\frac{1}{(5k+1)^{2}}$$ I mean, we can find the approximate value of it. But is there any chance to write this with known constants like ...
0
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0answers
38 views

Exponentiated Operators? ($e^{\hat{A}+\hat{B}} \ne e^{\hat{B}+\hat{A}}$)

Given, $$ e^{\hat{A}+\hat{B}} = e^{\hat{B}}e^{\hat{A}} $$ I then consider the series expansion of both exponentials. This then leads to a particular order of operation derived from the order of ...
1
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2answers
46 views

Prove that $\sum_{j=1}^{\infty} \frac{(-1)^{j+1} x^j}{j^2}$ is positive for $x \in (0,1)$

If $x \in (0,1)$, then $$\sum_{j=1}^{\infty} \frac{(-1)^{j+1} x^j}{j^2} >0. $$ How to prove this in an elementary way (without using properties of the polylogarithmic function $\text{Li}_2(-x) = - ...
2
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0answers
47 views

Does this series converge? If so, to what?

i was solving some integral equations and some of them gave series whose convergence am not very sure of. Problem, if anyone can point how or to what the series converges, i will be more than glad. Am ...
0
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2answers
76 views

Analytic continuation of $z-z^2+z^3-…$

I'm having trouble with the concept of analytic continuation of power series beyond the radius of convergence. For example for: $$f(z)=z-z^2+z^3-z^4+\cdots=\sum_{n=0}^\infty(-1)^nz^{n+1}$$ I get the ...
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
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4answers
48 views

Let $f(x)=\sum_{n=0}^{\infty}a_nx^n$ be convergent on $(-R,R),$ with $R>0.$ Prove that, if $f(x)=0$, $a_n=0$.

I'm stuck on a solution that our teacher gave to us. This is the exercise: Let $f(x)=\sum_{n=0}^{\infty}a_nx^n$ be convergent on $(-R,R),$ with $R>0.$ Suppose that $f(x)=0$ for all ...
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
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1answer
98 views

Get the closed form of Taylor series with Maple

Is it possible to get the closed form of Taylor series with Maple? The series command can give any given number of terms, but the question is about the closed form ...
2
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5answers
75 views

Radius of convergence of $ \sum_{n} \frac{x^n}{n\sqrt{n}}$

Trying to find the radius of convergence for $ \displaystyle \sum_{n} \frac{x^n}{n\sqrt{n}}$ I apply the root test: $\displaystyle \lim_{n \to \infty} ...
0
votes
1answer
43 views

Radius of convergence of $\sum_n \frac{(-1)^nx^n}{\ln(n+1)}$

Find the radius of convergence of the power series $\sum\limits_n \frac{(-1)^nx^n}{\ln(n+1)}$ $\displaystyle R = \frac{1}{\limsup\limits_{n \to \infty}\sqrt[n]{ ...
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 ...
3
votes
3answers
45 views

Differentiation of power series, problem

I have the power series $$u(x) = \sum_{k=1}^{\infty} \frac{x^{2k+1}}{k(2k+1)} $$ with radius of convergence $R \geq 1 $ and I want to perform termwise derivation for $|x| \lt 1$, but it isn't ...
1
vote
1answer
29 views

Laurent expansion on an annulus problem

I have $f(z)= \frac{1}{\sin(z)}$ and am required to show that on the disc {$0<|z|<\pi$} the Laurent expansion is equal to: $$c_{-1}z^{-1}+\sum_{n=0}^\infty{c_nz^n}$$ My plan is use the expansion ...
0
votes
1answer
47 views

Solving $(1-x^2)y''-2xy'+a(a+1)y=0$

I need to find an even solution and an odd solution to the ODE $(1-x^2)y''-2xy'+a(a+1)y=0$ using a power series around $x=0$. I suspect I should use Frobenius, but not sure how to bring it to the ...
53
votes
4answers
4k views

Factorial and exponential dual identities

There are two identities that have a seemingly dual correspondence: $$e^x = \sum_{n\ge0} {x^n\over n!}$$ and $$n! = \int_0^{\infty} {x^n\over e^x}\ dx.$$ Is there anything to this comparison? (I ...
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
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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 ...
1
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0answers
125 views

Infinitely nested radicals

In a recent paper it was stated (and maybe proved) that we can solve any polynomial equation with nested radicals. Here "nested radicals" means expression such as: $$ ...
0
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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 ...
1
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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 ...
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 ...
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 ...
0
votes
1answer
47 views

Sum of first $n$ coefficients of power series expansion

Consider the coefficients of the following expansion $$\frac{\left( e^{c t}-1 \right)^m}{(c \cdot t)^m e^{xt}}=\sum\limits_{n=0}^{\infty} A_n^m(x, c)\frac{t^n}{n!}.$$ Fix any $N,m,c\in\mathbb{N}.$ I ...
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 ...
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
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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
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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 ...
0
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1answer
22 views

The radius of convergence of a power series.

If I have a power series $$\sum_{j = 1}^{\infty}a_jx^{2j+1} = x\sum_{j = 1}^{\infty}a_jx^{2j} $$ Given that I have the radius of convergence $R$ of $$\sum_{j = 1}^{\infty}a_jz^{j}$$ where $z = ...
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 ...
0
votes
1answer
26 views

Meromorphic Function on Extended Plane

How do I prove that every meromorphic function on the extended plane is a rational function?
1
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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 ...
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
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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 ...
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 ...
0
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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 ...