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
26 views

proving convergence of a power series with the use of another power series

Let <$a_n$> be a sequence of nonnegative numbers such that $\sum_{n\ge1} a_{{_2}{^n}} $ converges.Prove that $\sum_{n\ge1} a_n $ converges. I tried but couldn't find a way to prove this.Any help ...
2
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3answers
102 views

find the radius of convergence of $(1+(-1)^{n}2^{1+n})x^{n}$

I ran into this question and I don't really know how to find the radius of convergence. the power series is: $$\sum_{n=0}^\infty(1+(-1)^{n}2^{1+n})x^{n}$$ Thanks in advance.
2
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1answer
71 views

Finding a power series for a peice-wise analytical fuunction.

Let $$f(x)=\begin{cases}x^2,& x\ge0,\\-x^2,& x\le 0.\end{cases}$$ Does there exist a power series $\sum_{n\ge 2} a_n x^n$ and a real number $R>0$ such that $\forall x\in(-R,R)$, ...
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2answers
85 views

$\sum^{\infty}_{n=1} \frac {x^n}{1+x^{2n}}$ interval of convergence

I need to find interval of convergence for this series: $$\sum^{\infty}_{n=1} \frac {x^n}{1+x^{2n}}$$ I noticed that if $x=0$ then series is convergent. Unfortunately, that’s it.
5
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0answers
204 views

How Ramanujan find this formula

I have seen this formula from Ramanujan $\sum_n \frac{a^{n+1}-b^{n+1}}{a-b}\frac{c^{n+1}-d^{n+1}}{c-d}T^n=\frac{1-abcdT^2}{(1-abT)(1-acT)(1-bcT)(1-bdT)}$. I know how to prove it via geometric ...
4
votes
3answers
198 views

How to prove that $\frac{1}{(1-x)^3}$ is the generating function for the triangular numbers?

How to prove that $\dfrac{1}{(1-x)^3}$ is the generating function for the triangular numbers? The $n^{\text{th}}$ triangular number is defined as $T_n = \displaystyle{n+1 \choose 2}$. I used ...
6
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6answers
420 views

Why is $ \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x$?

I am trying to see where this relationship comes from: $\displaystyle \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x$ Does anyone have any special knowledge that me and my summer math teacher doesn't know ...
0
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0answers
52 views

Proof of the power of this series?

Suppose: $$x = 1 + v + (1+ a + b) v^2/ 2! + (1+2a+b)(1+a+2b)v^3/3! + (1+3a + b)(1+2a +2b)(1+a+3b) v^4 /4! + \ldots$$ Prove that: $$x^n = 1 + nv + n(n+ a + b) v^2/ 2! + n(n+2a+b)(n+a+2b)v^3/3! + ...
1
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1answer
146 views

Trying to find more information about “Darboux's method/theorem” on coefficients of an analytic function

My supervisor briefly showed me a statement of something she called "Darboux's theorem," but I am having trouble finding more information about it on the internet. Here is what I have written down ...
6
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1answer
139 views

Recognizing if a power series is a $q$-expansion of a modular form

Given a power series in $q$, is it possible to tell if it is the $q$-expansion of a modular form (of level $N$ say)? I don't need to show results of this sort, but it has come up enough that I'm ...
1
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0answers
70 views

Are the special functions independent?

maybe the bessel functions are some complicated function of the exponential function, logarithm function... or maybe there's a relation between two or more transcendental functions. Is there a way to ...
1
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2answers
90 views

Convergence of $\sum_{n=0}^\infty z^{2^n}$

Let formally $f(z) := \sum_{n=0}^\infty z^{2^n}$. What is the raduis of convergence of this series ?
1
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1answer
61 views

Convergence of a complex power series

Let $a,b,c \in \mathbb C$ with $c \in \mathbb N$. Then I have to calculate the radius of convergence of the following power series: $$ 1+ \frac{ab}{c \cdot 1!} z + \frac{a (a+1)b(b+1)}{c(c+1)2!} z^2+ ...
1
vote
2answers
73 views

antiderivative of $\sum _{ n=0 }^{ \infty }{ (n+1){ x }^{ 2n+2 } } $

I've proven that the radius of convergence of $\sum _{ n=0 }^{ \infty }{ (n+1){ x }^{ 2n+2 } } $ is $R=1$, and that it doesn't converge at the edges. Now, I was told that this is the derivative of ...
2
votes
1answer
206 views

Why fractional power does not have power-series expansions?

Why fractional power does not have power series expansions? For example, $f(x)=x^{1/2}$, why the behavior at $0$ disallows a power-series expansion? For what reason? Thanks in advance.
2
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3answers
126 views

Explicit formula for the series $ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $

I was wondering if there is an explicit formulation for the series $$ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $$ It is evident that the converges for any $x \in \mathbb{R}$. Any ideas on a formula?
72
votes
3answers
2k views

Is the derivative the natural logarithm of the left-shift?

(Disclaimer: I'm a high school student, and my knowledge of mathematics extends only to some elementary high school calculus. I don't know if what I'm about to do is valid mathematics.) I noticed ...
1
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1answer
127 views

Is there a power series which pointwise convergent but not uniformly convergent on $(-1,1)$?

I was recently reading that power series of form $\sum_{n=0}^\infty b_n(x-a)^n$ converge uniformly to some uniform limit function on compact intervals $[a-r,a+r]$ if $r$ is less than the radius of ...
2
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0answers
100 views

Simplify the square of a sum of cosine functions

I have a square sum of exponantials as below: $$\left|\sum_{l=0}^{M-1}\exp\left(jl^2a\right)\,\exp\left(\frac{-j2\pi l}{M}b\right)\right|^2 $$ where $a$ is constant and $b$ is an integer . and I have ...
0
votes
1answer
474 views

Method for solving ODE with power series

when trying to solve second order linear homogeneous variable coefficient ODEs using a power series method, there seem to be two different general forms cropping up in my notes. The first uses an ...
5
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1answer
406 views

Maclaurin Series for $\ln(x+\sqrt{1+x^2})$

Is there a trick to finding the Maclaurin series for $f(x)=\ln(x+\sqrt{1+x^2})$ fast? Vaguely, I recall this being some sort of inverse hyperbolic function, but I'm not sure about which one, and what ...
2
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1answer
86 views

Showing a function is injective in a disk

Please help me prove that if $f(z)=\sum_{n=0}^\infty a_nz^n$ is analytic in $D_r=\{z:|z|<r\}$ and $r|a_1|>\sum_{n=2}^\infty n|a_n|r^n$, then $f$ is an injective function in $D_r$.
2
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4answers
125 views

How to compute $\sum_{n=0}^{\infty}\frac{(2n+1)}{n!}x^{2n+1}$?

I don't know how to compute $\sum_{n=0}^{\infty}\frac{(2n+1)}{n!}x^{2n+1}$,appreciate any help! Is there any general rule for solving such problems?
6
votes
1answer
106 views

Radius of convergence of $\sum_{n = 0}^{\infty} (a_1^n + \dots + a_k^n)z^n$, where $|a_1| = |a_2| = \dots = |a_k| = 1$

Here's the problem: Find the radius of convergence of $f(z) = \sum_{n = 0}^{\infty} (a_1^n + \dots + a_k^n)z^n$, where $|a_1| = |a_2| = \dots = |a_k| = 1$, and $a_i \in \mathbb{C}$. Since the series ...
11
votes
2answers
543 views

Taylor series (or equivalent at $\epsilon\to0$) of the integral over $x$ of a function of $x$ and $\epsilon$

I have a function $f$ of two arguments, defined as $$ f(x,\epsilon)=\epsilon\left( e^{-\frac{(x-\epsilon)^2}{2}} - e^{-\frac{x^2}{2}}\right) + \frac{1-\epsilon}{\sqrt{1+\epsilon}}\left( ...
1
vote
1answer
74 views

Proving that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$ with power series [duplicate]

Probably a simple question, but I wonder about the following: To prove that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$, I use : $$\exp(z_1+z_2) = ...
1
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0answers
84 views

Proof for closed form approximation of e

I am familiar with the derivation of $e$ from a power series, $e = \sum_{k=0}^{\infty} \frac{1}{k!} $ but have not found the proof for the following representation in any textbook $e = \lim_{x\ \to ...
10
votes
1answer
259 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 ...
0
votes
1answer
77 views

uniform convergence of $\sum\limits_{n=0}^{\infty} \frac {(-1)^nx^{2n+1}}{(2n+1)!}$

Given the series $\displaystyle \sum_{n=0}^{\infty} \frac {(-1)^nx^{2n+1}}{(2n+1)!}$ does the series converge on $\mathbb R$? I found that the radius is $\infty$ and I know that for $\forall c\in ...
1
vote
1answer
103 views

Necessary conditions for not having roots

Suppose $f(z)=\sum_0^\infty a_n z^n$ has a radius of convergence of $R$. What are necessary conditions, in terms of $\{a_n\}$, for $f(z)=0$ not to have any roots? Any combinations of real/complex ...
1
vote
2answers
426 views

Interval of Convergence of $\arctan(\frac{x}{\sqrt{2}})$

I am asked to find the power series of the function $f(x)=\arctan(\frac{x}{\sqrt{2}})$. I first found the derivative of this function which is: $f'(x)=\frac{\sqrt{2}}{2+x^{2}}$. Then I found the power ...
1
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0answers
25 views

Linearization of an expression involving some negative powers.

I have this adorable expression $$\left(R_1+i\ \omega\ (L_1+L_2)-i\frac{1}{\omega\ C_1}\right)+\left( \frac{1}{-i\frac{1}{\omega\ C_2}}+\frac{1}{R_3+R_{Gap}+i\ \omega\ L_3} \right)^{-1},$$ with ...
2
votes
2answers
57 views

A question on Power Series Radius Of convergence-need to check answer

What is the radius of convergence for the following power series I did some and want to check the answers $$ \sum\limits_{n=1}^\infty {n^2\over 2^n}x^{n^2} $$ I solved it the following way but I am ...
3
votes
0answers
157 views

Maclaurin's Series of Quotients and Products

As we all (should) know, the Maclaurin series is a special case of the Taylor series when the Taylor series is centered around 0. This is the canonical definition of the Maclaurin series: $$ f(x) = ...
2
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1answer
345 views

Uniform convergence of a power series

im new to this subject and very much appreciate your help with this Question, im not really sure about how to approach this, so if u can, please explain your steps $$f(x) = ...
5
votes
2answers
202 views

Interval of convergence of $\sum\limits_{n\geq0} \binom{2n}{n} x^n$

We consider the power series $\displaystyle{\sum_{n\geq0} {2n \choose n} x^n}$. By Ratio Test, the radius of convergence is easily shown to be $R=\frac{1}{4}$. For $x=\frac{1}{4}$, Stirling ...
6
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0answers
76 views

“Natural” interpolation between partial sums of a power series

Suppose $f(z)=\sum_{n=0}^\infty a_n z^n$ has a radius of convergence of $R$. Let the $N$-th partial sum be $f_N (z)=\sum_{n=0}^N a_n z^n$. What smooth (analytic) function interpolates between ...
2
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1answer
83 views

Find Taylor series expansion and convergence radius for $\int_0^x\cos(\sqrt{t}\ )dt$

i must find the the Taylor series expansion (i've been asked not necessarily calculating it directly) and the convergence radios for this function : $$f(x) = \int_0^x \cos(\sqrt{t}\ ) \, dt$$ I am ...
0
votes
3answers
70 views

Series Convergence of $1/(1+x)$

For what $x$ does $\sum_{k = 0}^{\infty} (-1)^k x^k$ converge (to $1/(1 + x)$) or diverge? Or does it converge within an interval like $\left[-1, +1\right]$?
4
votes
2answers
69 views

Interval of convergence for $\sum_{n=1}^∞({1 \over 1}+{1 \over 2}+\cdots+{1 \over n})x^n$

What is the interval of convergence for $\sum_{n=1}^∞({1 \over 1}+{1 \over 2}+\cdots+{1 \over n})x^n$? How do I calculate it? Sum of sum seems a bit problematic, and I'm not sure what rules apply for ...
4
votes
1answer
6k views

Multiplying two summations together exactly.

Consider the integral: $$\int_0^1 \frac{\sin(\pi x)}{1-x} dx$$ I want to do this via power series and obtain an exact solution. In power series, I have $$\int_0^1 \left( \sum_{n=0}^{\infty} (-1)^n ...
1
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2answers
500 views

What does analytic at a point means?

A function that is analytic at a point is one that can be represented by a Taylor or Maclaurin series? We also say that the radius of convergence should be positive. What if it was negative? What that ...
0
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1answer
153 views

Power Law Probability Distribution From Observations

This is probably a very simple question: I am trying to understand a power-law fitting technique by Aaron Clauset (http://tuvalu.santafe.edu/~aaronc/powerlaws/), but to do this I need to understand ...
5
votes
1answer
103 views

Proof that $\sum_{n=1}^{\infty} z^{1/n}$ doesn't converge

I believe I found a proof for the divergence of this sum for any value of $z$ besides 0. We can look on the telescopic series: $$\sum_{n=1}^{\infty}z^{1/(n+1)}-z^{1/n} = \lim_{N\rightarrow \infty} ...
4
votes
1answer
154 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 ...
0
votes
1answer
66 views

Calculate the Radius of convergence of $\sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$

I need your help: Calculate the Radius of convergence of the following: $$ \sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$$ Im new to this subject, so I'd appreciate it if you can add explanations to ...
3
votes
2answers
113 views

Change of a variable in a generating function

Assuming I have a generating function $$\sum_n c(m,n,k)x^n=\left(x\frac{1-x^m}{1-x~~~}\right)^k$$ (mentioned in this answer where $c$ represents the number of compositions of $n$ to $k$ parts of ...
9
votes
1answer
161 views

How to calculate the integral of $x^x$ between $0$ and $1$ using series? [duplicate]

How to calculate $\int_0^1 x^x\,dx$ using series? I read from a book that $$\int_0^1 x^x\,dx = 1-\frac{1}{2^2}+\frac{1}{3^3}+\dots+(-1)^n\frac{1}{(n+1)^{n+1}}+\cdots$$ but I can't prove it. Thanks in ...
1
vote
2answers
123 views

Which sets of positive rationals are closed under addition?

This question evolved because I was interested in generalizing power series so the exponents were rational numbers instead of integers, i.e., $\sum_{i=1}^{\infty} a_n x^{r_n}$, with the $a_i$ real and ...
1
vote
3answers
71 views

What's the formula for $\sum_{n=0}^{\infty}\left ( an+b \right )x^{n}$=?

Use the two formulas $\sum_{n=0}^{\infty}=\frac{1}{1-x}$ and $\sum_{n=0}^{\infty}\left ( n+1 \right )x^{n}=\frac{1}{(1-x)^{2}}$ to find a formula for this $\sum_{n=0}^{\infty}\left ( an+b \right ...