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).

learn more… | top users | synonyms

1
vote
1answer
40 views

Not quite alternating series

Quite a lot of things are known about alternating power series $$ \sum_{n \geq 0} (-1)^n a_n x^n, \quad a_n > 0, $$ like closed-form expressions for well-chosen $a_n$ and so on. In a problem I'm ...
1
vote
1answer
27 views

Equality between a fonction and an infinite serie.

I want to prove this equality : -ln(x+1) = S(from 1 to infinite) ((-x)^n/n) //for x between -1 and 1 (not included of course). I wrote on my notebook : show the derivates are equals and the two ...
1
vote
3answers
123 views

Radius of Convergence - $\sum_{n=1}^{\infty}2^n x^{n^2}$

What is the radius of convergence of this power series here? $$\sum_{n=1}^{\infty}2^n x^{n^2}$$
1
vote
1answer
1k views

Express the indefinite integral $\int\sin x^2~dx$ as a power series

What does this mean? I never saw this in my class/notes so I don't understand the conversion from integral to power series. Also if the integral were defined from $0$ to $1$, what new steps do I add? ...
0
votes
0answers
44 views

Is something wrong in this proof?

Show that if $\sum a_nx^n$ has convergence radio $R$ and $\limsup |a_n| > 0 $, then $R\leq 1$. Proof: Suppose that $\sum a_nx^n$ has convergence radio $R$ and $\limsup |a_n|=\alpha > 0$. ...
0
votes
1answer
61 views

Real Analysis: Proof of Radius of Convergence

I just would like to know are my steps right. $\textbf{I needed to show that the radius of convergence of $\sum_{n=0}^{\infty}a_nz^n$ is the same a that of $\sum_{n=0}^{\infty}na_nz^n$.}$
1
vote
1answer
61 views

Question about convergence of a power series and when the series is not zero

Following is a past exam question I am trying to solve as a preparation for my own exam. Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers with $a_n \leq M$ for some $M\in\mathbb{R}^{+}$ and ...
1
vote
3answers
69 views

$S(t) = \sum_{n=1}^{+\infty}\frac{e^{int}}{n}$

Let $t \in (0, 2\pi)$, calculate $S(t) = \sum_{n=1}^{+\infty}\frac{e^{int}}{n}$. If we can derive term by term is easy, but how to prove that it is possible ?
3
votes
1answer
42 views

$\sum_{n=0}^{+\infty}|a_{n}-a_{n+1}| $ converges $\Rightarrow \sum_{n=0}^{+\infty}a_{n}z^{n}$ converges

Let $\{a\}_{n} \subset \mathbb{C}$ , $a_{n} \rightarrow 0 $ , $\sum_{n=0}^{+\infty}|a_{n}-a_{n+1}| < +\infty$; then show that $\sum_{n=0}^{+\infty}a_{n}z^{n}$ converges if $|z|\le 1 \ $ , $z \ne ...
2
votes
2answers
59 views

Power series of trigonometric functions

Problem statement: Determine those $x$, for which the power series is convergent and determine the sum. $$f(x)=x+\sum_{n=2}^{\infty}(-1)^{n-1}2n\frac{x^{2n-1}}{(2n-1)!}$$. Progress: I have ...
0
votes
3answers
71 views

Prove the Identity For Fringe Patterns

Prove the two Identities for $-1 < r < 1$ $$\sum_{n=0}^{\infty} r^n\cos n\theta =\frac{1-r\cos\theta}{1-2r\cos\theta+r^2}$$ $$\sum_{n=0}^{\infty} r^n\sin{n\theta}=\frac{r \sin\theta ...
2
votes
3answers
54 views

A power series from $\frac{x}{9+x^2}$

I need to make power series from $\frac{x}{9+x^2}$, and I don't have any idea how. The only thing I know is how to make power series from $\frac{1}{1-q}$. Thank you!
1
vote
2answers
30 views

Help with power series $f(x)=\frac{2x}{(1-x^2)^2}=\frac{d}{dx}\left(\frac{1}{1-x^2}\right)$

Given that $f(x)=\frac{2x}{(1-x^2)^2}=\frac{d}{dx}\left(\frac{1}{1-x^2}\right)$, find a power series for $f(x)$. What is its radius of convergence? So far I got the following: ...
1
vote
2answers
187 views

Uniform convergence of the series for $\frac{x}{\sqrt{1+x}} = \sum_{n=1}^\infty (-1)^n\binom{-1/2}{n}\left(\frac{x}{1+x}\right)^{n+1}$

I am looking for the values where this series expansion converges uniformly. $$\frac{x}{\sqrt{1+x}} = \sum_{n=1}^\infty (-1)^n\binom{-1/2}{n}\left(\frac{x}{1+x}\right)^{n+1}$$ Intuitively, I believe ...
5
votes
3answers
383 views

Radius of convergence and the endpoints of a power series

Find the radius of convergence and the convergence at the end points of the series: $$\sum_{n=1}^\infty(2+(-1)^n)^nx^n$$ This is what I did: $a_n=(2+(-1)^n)^n\Rightarrow ...
5
votes
1answer
100 views

Find a power series for the following function $\frac{x^2}{1+x^3}$

I'm not sure if I solved this correctly, but here is the problem: Find a power series for the following function $\frac{x^2}{1+x^3}$ And here is what I did: $$x^2\frac{1}{1-(-x^3)}$$ Here is where ...
4
votes
3answers
260 views

Asymptotic expansion of $J(t) = \int^{\infty}_{0}{\exp(-t(x + 4/(x+1)))}\, dx$

I want to derive an asymptotic expansion for the following Bessel function. I think I need to rewrite it in another form, from which I can integrate it by parts. I am interested in obtaining the ...
1
vote
0answers
1k views

Taylor's Formula vs. Taylor's Inequality

In my calculus book, Essential Calculus, and in class we were using Taylor's formula to approximate the remainder in Taylor polynomials but I am having a bit of trouble understanding the intuition ...
0
votes
1answer
80 views

radius of convergence of the power series $\sum_{n=0}^{\infty} z^{n!}$

How to find the radius of convergence of the power series $$\sum_{n=0}^{\infty} z^{n!}?$$ I don't know how to start !!!
0
votes
1answer
359 views

Finding a solution in powers of $(x^2-1)y''+4xy'+2y=0$

I'm working on finding the general solution of $(x^2-1)y''+4xy'+2y=0$ in powers. I assume the form: $$ y(x)=\sum_{n=0}^\infty C_nx^n$$ My basic strategy is to first figure out each piece ...
0
votes
1answer
71 views

Determine the sequence of coefficients $(a_n)_{n\in\mathbb{N_0}}$ so that: $\sum_{n=0}^\infty a_nx^n = \frac{e^x}{1-x} $

Assignment: Determine the sequence of coefficients $(a_n)_{n\in\mathbb{N_0}}$ so that $$\sum_{n=0}^\infty a_nx^n = \frac{e^x}{1-x}\ , \forall x\in\mathbb{R}: |x| < 1. $$ What I've got so far ...
1
vote
1answer
58 views

Determine the radius of convergence of the power series $\sum_{k=1}^\infty \frac{1-(-2)^{(-k-1)}k!}{k!} (z-2)^k$.

I could use some help solving this one. Applying the nth-root or the ratio test didn't work out for me. $$\sum_{k=1}^\infty \frac{1-(-2)^{(-k-1)}k!}{k!} (z-2)^k$$ Hints are just as appreciated ...
1
vote
0answers
60 views

Taylor's Formula and 'z' values

I have attached Taylor's Formula, an exercise problem from a section on Taylor polynomials, and the solution to this exercise. I understand part a, expanding $f$ using Taylor polynomials is the ...
6
votes
4answers
643 views

If $\sum_{n = 1}^\infty {{a_n}}$ converges, then is $\sum_{n = 1}^\infty (1+a_n)^{-1}$ a convergent series?

If $\sum\limits_{n = 1}^\infty {{a_n}}$ is convergent (with ${a_n} > 0$, $\forall n\in\mathbb{Z}$), then is $\sum\limits_{n = 1}^\infty {\left( {\frac{1}{{1 + {a_n}}}} \right)}$ is a convergent ...
2
votes
1answer
31 views

Calculate $\sum_{n=m}^\infty (-1)^n \frac{(a)_n}{(n-m)!}x^n$

Can you find an analytical expression for the following series? $$\sum_{n=m}^\infty (-1)^n \frac{(a)_n}{(n-m)!}x^n$$ where $m$ is a nonnegative integer, $x\in (0,1)$, $a > 0$, and $(a)_n$ is the ...
1
vote
1answer
54 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
1answer
215 views

How to prove that a complex power series is differentiable

I am always using the following result but I do not know why it is true. So: How to prove the following statement: Suppose the complex power series $\sum_{n = 0}^\infty a_n(z-z_0)^n$ has radius of ...
0
votes
2answers
40 views

Prove something about power-series

Lets say that for every $n\in \mathbb{N}$$,a_n\ne0$. If, $$\lim_{n\to 0}\frac{a_{n+1}}{a_n}=l$$ how do I prove that the radius of the convergence of the series:$$\sum_{n=0}^{\infty}a_nx^n$$ is ...
1
vote
1answer
45 views

For what $m,\ k$ values does this series converge?

$\sum_{n=1}^{\infty}{\sqrt[m]{n!}\over{\sqrt[k]{(2n!)}}}$ I am inclined to think that factorial type questions are best solved with the ratio test. I was able to reach the following: ...
1
vote
3answers
39 views

Alternating sign for unequal numbers

How to model a function which satisfies following condition: $$ f(x) = \begin{cases} 1 & x \in 3,7,11,\ldots\\ -1 & x \in 1,5,9,\ldots \end{cases} $$ The first result can be ...
0
votes
1answer
46 views

Sum $\sum_{n=2}^\infty\frac{a^{n+1}}{n(n-1)}z^{n}$

Given the power series $\sum_{n=2}^\infty\frac{a^{n+1}}{n(n-1)}z^{n}$ , where $a>0$, find the radius of convergence and the sum of the series. The radius is $\frac{1}{a}$ , but what about the sum? ...
1
vote
0answers
131 views

Power series to calculate LambertW up to infinity?

Is this an allowed operation to calculate the Lambert W function as a power series up to infinity, or is there some trouble in defining it this way? Mathematica programs: ...
1
vote
1answer
147 views

problem with recurrence relation for series solution for ODE

I have $$y''-xy'-y=0$$ and I'm trying to find the series solution around the ordinary point $x_0=1$. My last post I muscled through to the solution when the ordinary point was $x_0=0$, but this is ...
0
votes
1answer
40 views

Complex power series (or not quite so?)

I'm stuck with this problem. Any hints are appreciated. It just says $$ \mbox{"For what values of}\ z\ \mbox{is}\quad \sum_{n = 0}^{\infty}\left(z \over 1+z\right)^{n}\quad \mbox{convergent ?} $$ ...
1
vote
0answers
96 views

Are there other power series for the Lambert W function than this one?

Are there other known power series for the Lambert W function, other than this one: $$W(x) = x-x^2+\frac{3 x^3}{2}-\frac{8 x^4}{3}+\frac{125 x^5}{24}-\frac{54 x^6}{5}+\frac{16807 ...
2
votes
0answers
182 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 ...
1
vote
2answers
192 views

Why is the circle of convergence for complex power series a circle (and not e.g. a square)?

Power-Series have an "circle of convergence". With real numbers this is an interval. Expanding this to complex numbers this becomes a circle. There are lots of book stating this, but I did not find ...
2
votes
1answer
94 views

sum of an alternating series

How to evaluate the series below ? $$ \sum_{n=0}^{\infty}\left(-1\right)^{n}\,{2n+1 \over \left(2n+1\right)^{2} + x^{2}} $$ Can we reexpress it in term of an elementary function ?.$\,$ By the way, ...
1
vote
1answer
211 views

Negative Base to non-integer power

I'm looking to consistently solve the m^n case, including conditions where m is negative and n is non-integer. I'd like to, additionally, catch the error when it isn't possible. Some examples to ...
3
votes
0answers
80 views

Is there a gap in Serre's proof of inverse function theorem?

On page 73 of 'Lie algebras and Lie groups', Serre proves the inverse function theorem for complete fields. I would like to have some clarification about the following point. Let $K$ be a complete ...
1
vote
0answers
93 views

Radius of convergence for Taylor series?!

Given is: $f(x) = \frac{\sin x}{x} $ I need the Taylor series in $a = 0$, so: $$T(x,0) = \frac{1}{x} \sum_{n=0}^\infty ((-1)^n* \frac{x^{2n+1}}{(2n+1)!} ) = \sum_{n=0}^\infty (-1)^n * ...
0
votes
1answer
39 views

Using differentiation to find a power series representation of the following function

The problem that was given was to use differentiation to find a power series representation of the following function $\frac{1}{(x+6)^2}$. I know how to find the power series representations of ...
1
vote
1answer
37 views

Interval of convergence homework

$$ \sum_{n=1}^{\infty} \left(\frac{-1}{4}\right)^n \frac{(5n)^n}{n!} (x-1)^n $$ First, I start with the ratio test: $$\lim_{n\to \infty} \left| \left(\frac{-1}{4}\right)^{n+1} ...
0
votes
1answer
115 views

Find the sum of $\sum (n^2+n)x^n$ using integrals

I'm having a difficult to find $\sum_{n=1}^\infty (n^2+n)x^n$. the solution is $\frac{2x}{(1-x)^3}$. This is my solution: $$1. \space\space\space\space S(x) = \sum_{n=1}^\infty (n^2 +n)x^n =$$ ...
0
votes
3answers
230 views

Series Expansion of $\arcsin\left(\frac{a}{a+x}\right)$

Can anyone think of a good approximation to: $$ \arcsin\left(\frac{a}{a+x}\right)\ $$ accurate at $x = 0$? The Taylor series is not available...perhaps some other kind of method?
1
vote
0answers
50 views

Where on the border of convergence circle series converges and where diverges?

I have power series of $ \sum\limits_{k=2}^{\infty} (\ln k)^{\alpha} z^k$. Alpha is a parameter. I've found the radius of convergence. R = 1. If $alpha \geq 0$ then series diverges for z from boundary ...
0
votes
1answer
63 views

$\log(1-x)=\sum_{n=1}^{\infty}{-\frac{x^n}{n}}$

I want to show that the power series around $0$ corresponding to the function $f:x\mapsto \log(1-x)$ is $\sum_{n=1}^{\infty}{-\frac{x^n}{n}}$. I know that the series $\sum_{n\ge 1}{-\frac{x^n}{n}}$ ...
1
vote
1answer
106 views

Is there a power series which converges to $f(x) =| x|$ for all $x$?

I'm confused how to solve the following problem: "Is there a power series which converges to $f(x)$ = $\left| x\right|$ for all $x$?" Your help is greatly appreciated. Thanks a lot!
2
votes
1answer
115 views

Choice of the First Term in Legendre Polynomials

The two solutions of the Legendre's Differential Equation obtained by series solution method are : and Now according to my textbook, for the useful polynomial for n equal to a positive integer, ...
1
vote
0answers
60 views

Equality between sum and integral

Show that,(where log is natural logarithm) $\lim_{n\rightarrow\infty}\left(logn\ -\sum_{k=0}^n\frac 1k\right)=\lim_{n\rightarrow\infty}\int\limits_{0}^{n}\left(1-\dfrac xn\right)^nlogx\ dx$ ...