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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1answer
29 views

Looking to have my worked check on a calculus series question

I am trying to determine if the Taylor series of $f(y) = y^{-\frac{1}{3}}$ about $y=1$ converges absolutely at $y = 2$. I am calculating the Taylor series as $$f(y) = 1 + a_1 (y-1) + a_2 (y-1)^2 + ...
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1answer
22 views

Two convergent power series are the same if they equal on an infinite set of points having 0 as a limit point.

I'm having difficulty following the proof of the theorem below. First of all, how do we know that h(z) is a power series having a non-zero radius of convergence from the fact that f(z) is. And, ...
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1answer
12 views

Factoring to find Power Series and Radius of Convergence

The question asks: Find a Power Series representation and Radius of Convergence for: $f(x)=\dfrac{x}{9+x^2}$ I see that it is pretty straight forward that if I re-write this by factoring out a 9, ...
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0answers
22 views

Convolution vs term wise product

Fix an element $x\in\mathbb R^\times$. Then I can define the ring $\mathbb R[[x]]$ which is the set: $$\left\{\sum_{i=0}^\infty a_ix^i\,: a_i\in\mathbb R\right\}$$ and where the product is the usual ...
2
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1answer
30 views

Complex power series and radius of convergence

Let $c$ be a non-zero complex number, and consider the power series \begin{equation} S(z)=\frac{z-c}{c}-\frac{(z-c)^2}{2c^2}+\frac{(z-c)^3}{3c^3}-\ldots. \end{equation} By using the Ratio Test, or ...
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2answers
49 views

Finding Laurent's series of a function

I am trying express the function $$f(z)=\frac{z^3+2}{(z-1)(z-2)}$$ like a Laurent's series in each ring centering in $0$, but I do not now how could I express it, in first I said that ...
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1answer
94 views

Statistical problem: how many books of different widths fit it into a self of a limited certain width?

Let's say I have N sets of books, being the size of the books in a set the same. The cardinality of the every set is different: I might have 3 books of width 5 units (first set), 6 books of width 10 ...
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2answers
73 views

What is a power series representation of $\frac{1-x^2}{1+x^2}$

I got so far as to rewrite to this: $(1-x^2)\cdot \frac{1}{1-(-x^2)}$ so that I can write the power series as follows: $$(1-x^2)\cdot\sum_{n=0}^\infty(-1)^n \cdot x^{2n}$$ But how can I bring the ...
3
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1answer
33 views

If $\sum a_n z^n$ has $r \gt 0$ then there is a $C \gt 0$ such that if $A \gt 1/r$ then $|a_n| \le CA^n$

Suppose that $\sum a_n z^n$ has a radius of convergence greater than $0$. Then there exists a positive number $C$ such that if $A \gt 1/r$ (where $r$ is the radius of convergence) then $$|a_n|\le ...
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0answers
23 views

Hadamard Product for Series.

We know that Hadamard product for two series $\displaystyle A=\sum_{n=1}^\infty a_nx^n$ and $\displaystyle B=\sum_{n=1}^\infty b_nx^n$ is $\displaystyle A*B=\sum_{n=1}^\infty a_nb_nx^n$. If every case ...
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0answers
58 views

Geometry of the zeros of a power series.

This is probably a basic question that is easily googlable, but it seems that I dont have the right keywords. So my question is, having some power series $$ f(z)=\sum_{k=0}^{\infty}C_{k}z^{k}, ...
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1answer
47 views

Sum of the power series $\sum\limits_{n=2}^{\infty}\frac{n^2}{(n-1)(n+2)}x^n$

The series is convergent for $|x|<1$ and divergent for $|x|>1$. I can't find the sum. Integrating three times gives $$\frac{n^2}{(n^2-1)(n+3)(n+2)^2}x^{n+3}$$ that should have a closed form. ...
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2answers
44 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 ...
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1answer
24 views

Sine squared when obtaining a Maclaurin and taking it's limit as $x\to 0$

When determining the Maclaurin series for $\sin^2(x)$ we use the trigonometric identity $\frac{1-\cos(2x)}{2}$. But when taking the limit of e.g. the Maclaurin series for $\cos(\sin(x))$ as $x$ ...
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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 ...
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1answer
32 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 ...
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2answers
35 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 ...
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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 ...
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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 ...
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1answer
20 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 ...
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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 ...
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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 ...
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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 ...
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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) = - ...
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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 ...
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4answers
434 views

How to find the sum of the infinite series whose general term is not easy to visualize

I am to find out the sum of infinite series:- $$\frac{1}{6}+\frac{5}{6\cdot12}+\frac{5\cdot8}{6\cdot12\cdot18}+\frac{5\cdot8\cdot11}{6\cdot12\cdot18\cdot24}+...............$$ I can not figure out the ...
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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. ...
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2answers
75 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 ...
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1answer
450 views

Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$?

This was inspired by this post. Let $q = e^{2\pi\,i\tau}$. Then, $$x := \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$ ...
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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 ...
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1answer
61 views

What is the significance of this identity relating to partitions?

I was watching a talk given by Prof. Richard Kenyon of Brown University, and I was confused by an equation briefly displayed at the bottom of one slide at 15:05 in the video. $$1 + x + x^3 + x^6 + ...
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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]{ ...
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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} ...
3
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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 ...
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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 ...
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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 ...
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1answer
90 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 ...
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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 ...
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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
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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 ...
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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 ...
4
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1answer
168 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 ...
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0answers
35 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
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2answers
99 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
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0answers
29 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 ...
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2answers
25 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
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1answer
40 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
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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
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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 $(-1,1)\times(0,\infty)$ $\mathbb R\times(-1,1)$ ...