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
137 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 ...
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1answer
40 views

Complex power series (or not quite so?)

I'm stuck with this problem. Any hints are appreciated. It just says $$ \mbox{"For what values of}\ z\ \mbox{is}\quad \sum_{n = 0}^{\infty}\left(z \over 1+z\right)^{n}\quad \mbox{convergent ?} $$ ...
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0answers
90 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
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0answers
165 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 ...
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2answers
171 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
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1answer
89 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, ...
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1answer
187 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
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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
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0answers
82 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 * ...
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1answer
36 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 ...
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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} ...
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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: $$1. \space\space\space\space S(x) = \sum_{n=1}^\infty (n^2 +n)x^n =$$ ...
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3answers
204 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?
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0answers
49 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 ...
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1answer
62 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
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1answer
105 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
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1answer
111 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
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0answers
59 views

Equality between sum and integral

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

solution needed to modify power series

I want to find the power series for $\sqrt{x}$. I surf on the internet but there is no success. I found some link from that I created power series of $\sqrt{x}$ as below ? The link I refer is ...
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2answers
47 views

Power series representation for $\frac{3}{(1-x)^2}$

I integrated to get $\frac{3}{1-x}$, turned it into a power series $3x^n$, and differentiated to get the series $3nx^{n-1}$ which is incorrect.
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1answer
42 views

Can any one tell me the books for power series?

Can any one tell me the books for power series? I want to find the power series for sqrt(x). I surf on the internet but there is no success. So please tell me the name of the book where I can find ...
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2answers
116 views

Values of $k$ for non-trivial solutions of the differential equation $y''-\left(\frac{1}{4}+\frac{k}{x}\right)y=0$ where $x$ is non-negative

I attempted a power series solution of this equation in order to find the values of k that have a non-trivial solution: $y''-\left(\dfrac{1}{4}+\dfrac{k}{x}\right)y=0$ I am having trouble ...
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1answer
76 views

Is $k(X)\subset k((X))$ an algebraic extension of fields?

Let $k$ be a field, and consider the field $K=k((X))=\text{Frac}(k[[X]])$. Then there exist a transcendence basis $\mathscr{B}$, i.e., a subset of algebraically independent element of $K$ such that ...
3
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1answer
75 views

Find the sum of $\sum_{n=1}^\infty (-1)^{n+1} (2n-1)x^{2n-1}$

Find the sum of $S(x) = \sum_{n=1}^\infty (-1)^{n+1} (2n-1)x^{2n-1}$. I know convergence radius is $1$ because $\frac{1}{\sqrt[n] {(2n-1)}} = \frac{1}{1} = 1.$ Then: $$x^{-1} S(x) = ...
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3answers
200 views

Find a power series solution centered at 0 (Differential equations

Here's the problem: $$(x-1)y''+y'=0$$ This is the work that I've already done: $$y=\sum_{n=0}^{\infty}a_{n}x^n$$ $$y'=\sum_{n=0}^{\infty}(a_{n+1})(n+1)x^n$$ ...
0
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1answer
79 views

Convergence of formal power series substitution

Prove that the substitution of formal power series $F(G(x))=\sum_{k\geq0}f_k \frac{G(x)^k}{n!}$ converges for every $F$ if and only if $G(0)=0$
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1answer
50 views

Sum of the series $\sum_{n \ge 0}{\frac{x^{4n+1}}{(4n+1)!}}$

I want to determine the sum of the series $$\sum_{n \ge 0}{\frac{x^{4n+1}}{(4n+1)!}}$$ I know this has to do with the sum $$\sum_{n \ge 0}{\frac{x^{n}}{(n)!}}=e^x\;\; \forall x\in \mathbb R$$ But i ...
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1answer
137 views

Question about the infinite products of formal power series

I need a proof for this: Let $(F_j)_{j\ge 0}$ be a sequence of formal power series. The infinite product $\prod_{j\geq0}(1+F_j(x))$, where $F_j(0)=0$, converges if and only if ...
1
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1answer
176 views

Radius of convergence and sum of $\sum_{n\ge 0}{\frac{\cos(n)x^n}{n!}}$

I want to determine the radius of convergence of the power series $$\sum_{n\ge 0}{\dfrac{\cos(n)x^n}{n!}}$$ The ratio test gives that for $x\not = 0$ we have that ...
3
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2answers
672 views

Formal power series, the Chain Rule and the Product Rule.

Definitons Let $$\mathbb{C}[[x]] := \left\{ \sum_{n\geq 0} a_n x^n : a_n \in \mathbb{C} \right\}$$ be the set of formal power series of $x$. Exercise i) If $F_1(x)$ and $F_2(x)$ are power series ...
2
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1answer
147 views

The sum of the power series $\sum_{n\ge 0}{\frac{(n-1)x^n}{n+1}}$

I want to determine the sum of the power series $$\sum_{n\ge 0}{\frac{(n-1)x^n}{n+1}}$$ Is my try correct ? I know it is not correct since the sum I get is not continuous at $x=0$ but I can't find my ...
1
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1answer
187 views

Find the sum of $\sum_{n=1}^\infty \frac{x^n}{2^{n+1}}$

I need to find the sum of $\sum_{n=1}^\infty \frac{x^n}{2^{n+1}}$, But I don't really know how. I tried to manipulate integrals and derivetives but it's not helped me. Can you pleae help me to find ...
1
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0answers
56 views

Radius of convergence of $\sum_{n\ge 1}{\frac{x^{n^2}}{n^2}}$

I want to determine the radius of convergence of the power series $$\sum_{n\ge 1}{\frac{x^{n^2}}{n^2}}$$ Is my following try correct, and is there any simpler way to do this: Put ...
2
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0answers
41 views

Power Series for Original Differential Equation

The question: $y"+x^2y'+2xy=0$ I continue to get the incorrect answer and not sure why. I changed my indices around to make x^n all throughout and that's where the trouble starts. My answer ...
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2answers
151 views

The radius of convergence of $\sum_{n\ge 0}{\log(n!)x^n}$.

I want the radius of convergence of the series $\sum_{n\ge 0}{\log(n!)x^n}$. Could I use the stirling formula $$n!\sim_\infty \left(\frac{n}{e}\right)^n\sqrt{2 \pi n}?$$ Because then $$\log ...
1
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1answer
104 views

Radius of convergence of $\sum_{n\ge 1}{n^{\log(n)}}x^n$

I want to find the radius of convergence of the power series $\sum_{n\geq 1}{n^{\log(n)}}x^n$. Is my following try correct and is there any other simpler way to do this: Let $a_n(x)=n^{\log(n)}x^n$. ...
1
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0answers
195 views

Second Series solution y(2) for Frobenius Method

I am currently solving the Frobenius Method for the question $xy'' +y = 0$ given the ICs $y(0) = 0, y'(0)=1$ I have done some work into solving that the first series solution for $y_1 = ...
2
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1answer
214 views

The Lagrange inversion formula (the proof from Wikipedia)

English Wikipedia gives a very short proof of the Lagrange Inversion Theorem, using the formal residue. However, I don't understand the second equality, $$k \, \mathrm{Res} (g^n \, X^{-k-1}) = k \, ...
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3answers
219 views

2013th derivative of rational function

I am struggling to find $f^{(2013)}(0)$ for $$f(x) = \frac{1}{1 + x + x^3 + x^4}$$ I know that I should use power series, and following a hint I rewrote the equation as the following: $$1 = (1 + x + ...
2
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1answer
69 views

power series method to solve ODE

Using power series method, solve Airy’s equation $$y′′+ xy = 0$$. How do I start solving this? Thanks in advance!
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1answer
85 views

rewrite a geometric series as a power series?

Is there an easy way to rewrite the series $$\sum_{n=1}^\infty a_n = \frac{81x^3}{(81x^4-1)^2}$$ as a power series?
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1answer
60 views

generating function of a sequence

There are n lines drawn in a plane such that no 2 lines are parallel and no 3 lines are concurrent. If the plane is then divided into an regions prove that a1=2 a2=4 a(n)=a(n-1)+n for n>=2 Find the ...
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1answer
743 views

What are some practical uses of power series?

Why do we care about power series? Why are they important?
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1answer
142 views

Differential Equations: Find the first four terms in each of two solutions y1 and y2 …

The differential equation is $y'' - xy' - y = 0$ with $x_0 = 1$ Now, I know how to find the recurrence relation... and it's given by: $a_(n+2) = [(a_(n+1) + a_(n)) / (n+2)]$ But I can't quite ...
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2answers
36 views

Geometric series - regarding the exponents of the factor r

When giving a proof for the formula of a geometric sum, the following sequence is used $$S = a + ar + ar^2 + ... + ar^{n-1}$$, Why does the last term go to $n-1$ and not just $n$? I wrote the last ...
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3answers
38 views

Finding the interval of convergence

I need to find the interval of convergence of the power series $$\sum_{n=1}^\infty (-1)^n\frac {{n^2}{x^n}}{2^n}$$ I have deduced that the radius of convergence is $1$, but am unsure of how to then ...
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2answers
63 views

Multiplicative Inverse for Generating Function

I have a question based on Irreducible and Connected Permutations. I was able to use the notion of connected permutations to construct a combinatoric proof for \begin{equation} ...
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1answer
142 views

Bound of power series coefficients of a growth-order-one entire function

An entire function $f(z)$ satisfies $$|f(z)| \leq A_\varepsilon e^{2\pi(M+\varepsilon)|z|}$$ for every positive $\varepsilon$. I want to show that $$\limsup_{n \to \infty}\ [f^{(n)}(0)]^{1/n} \leq ...
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1answer
40 views

Power Series Expansion Asymptotics

From my text: Given $\cos^n(x),$ set $x=\frac{\omega}{\sqrt{n}}$, then a local expansion yields: $\displaystyle\cos^n(x)=e^{n\log\cos(x)}=e^{(-\frac{\omega^2}{2}+O(n^{-1} \omega^4))}$ ...
1
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0answers
105 views

How can we interpret the coefficients of Laurent series?

The coefficients of a Taylor series of a function about a given point are related to the nth derivatives of the function at that point. Can we make a similar statement about what the (negative-index) ...