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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0answers
24 views

Find behavior near fixed point beyond linear expansion

this is my first question on math.stackexchange, I hope to have phrased it correctly! I have a differential equation $\text{$\frac{\text{d}x}{\text{d}t} = \alpha t^{-3}\frac{f'(x)}{f(x)}$ with ...
2
votes
2answers
50 views

What would be a power series for $f(z)=\sin(z)$ centered at $1$?

Everything is in the question! I've seen loads examples like "centered at $\pi$, $\pi/2$,... But $1$ would make everything much different... I've tried to work this way: $\sin(z) = \sin((z-1)+1) = ...
0
votes
1answer
23 views

Find the series solution for the ODE $x^2y''(x)-3y(x) = 0$

Find the series solution for the ODE $x^2y''(x)-3y(x) = 0$ I assume $y(x) = \sum{a_nx^n}$ then substitute in the equation and get $$\sum_{n=0}^{\infty} ({a_nn(n-1) - 3a_n)x^n}=0$$ When I ...
2
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1answer
40 views

prove that a function is expressible as a power series

I started by rearrange f(z), and expanded the terms in summation. Then, I did not get very far. It would be great if anyone can let me know what is needed to figure out bn. Thanks in advance.
1
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0answers
34 views

asymptotic of a power series

Show that \begin{equation*} \sum_{n=0}^{+\infty}x^{n^{2}} \end{equation*} is equivalent to \begin{equation*} \frac{1}{2}\sqrt{\frac{\pi}{1-x}} \end{equation*} as $x\in (0,1)$ approaches $1$. This ...
0
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2answers
36 views

How to solve differential equation using power series?

$$x^3y'' + xy′ + 2y = 0 $$ Find a number $r \in \mathbb{R}$ and coefficients $a_n$ such that $y(x) = x^r \sum_{n=0}^{\infty}a_n x^n$ is a non-constant solution of the equation above. I am having ...
2
votes
1answer
18 views

How to prove coefficients of a power series is bounded?

Let $f(z)=\sqrt{1-z}$. Let $$\sum_{k=0}^\infty c_kz^k$$ be the power series converges to $f(z)$ in the ball $|z|<1$. How can I prove that $|c_k|$ is bounded.
1
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2answers
25 views

Square root of a bounded operator in Hilbert space

Consider the power series expansion $$ \sqrt{1-z} = 1+\sum\limits_{k=1}^\infty c_k z^k, $$ converging absolutely in the ball $|z| \leq 1$. Let $H$ be a Hilbert space and $A \in \mathcal L(H)$ a ...
2
votes
1answer
105 views

summation of a finite sequence?

What is the summation of the finite sequence: $$\sum\limits_{i = 1}^n {\frac{1}{i}\left( {\begin{array}{*{20}{c}} {2i - 2}\\ {i - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {2n + 2 - 2i}\\ ...
5
votes
1answer
123 views

what is the summation of such a finite sequence?

The summation is: $$\sum_{i=0}^n \binom{2i}i \binom{2n-2i}{n-i}$$ The answer is $4^n$. How to prove it, and how to think out it?
1
vote
1answer
38 views

taking the inverse of power series

I am working with solution to near regular singular points. I started with: $$y_1(x)=x^\frac{1}{2}\left[1-\frac{3}{4}x+\frac{9}{64}x^2-\frac{3}{256}x^3+\cdots\right] $$ Then I squared it: ...
0
votes
1answer
50 views

Exponentiation a power series

I find formula to calculate $f(z)=(\sum_{i=0}^{\infty} {x^{2i+1}})^n.$ I know that $\sum_{i=0}^{\infty} {x^{2i+1}}=\frac {x}{1-x^2}.$ But I need function $f(x)$ as a power series. This is my ...
3
votes
1answer
36 views

Find complex power series expansion for $\int e^{-w^2} dw$

If a function $E(z)$ is defined on $\mathbb{C}$ by $$ E(z) = \int_0^z e^{-w^2} dw,$$ find a power series expansion for $E(z)$ about $0$. What does this power series converge? I know how this ...
0
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0answers
12 views

Asymptotic power series of $F(x)= \int_{0}^{T} f(t) e^{-xt}dt $

Let $f \in C^{\infty}$ and $T>0$. I am asked to find an asymptotic power series of the funtion: $$ F: \mathbb{R}_{+} \rightarrow \mathbb{R}, \quad F(x)= \int_{0}^{T} f(t) e^{-xt}dt$$ as $x ...
0
votes
1answer
25 views

Contour integral of convergent power series

Given that $\frac{e^z}{z^k} = z^{-k} + z^{1-k} + \frac{z^{2-k}}{2!} + \frac{z^{3-k}}{3!} + ...$ converges uniformly on any set $\{z \in C: r \leq |z| \leq Z\}$ (where $0 < r < R$), show that for ...
5
votes
1answer
35 views

Solving 2nd order ODE with Frobenius method - problems with summation symbol

I'm trying to solve the ODE: $$ y''(x) + \frac{2x}{(x-1)(2x-1)} y'(x) - \frac{2}{(x-1)(2x-1)} y(x) = 0 $$ I'm trying to find a solution by the Frobenius method, expanding a power series of the ...
2
votes
1answer
37 views

Which solution is the right one??

If we want to solve the equation $sec^2(x)$ for finding the all roots(real and complex), we have two ways: 1-Direct solving for $sec^2(x)=0$ 2-Or by convert the above equation to polynomial series as ...
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2answers
37 views

Is $-\log (1-z) = \sum_{n=1}^{\infty}\frac{z^n}{n}$ for $z \in \mathbb{C}, \|z\|=1, z \neq 1$?

Is $-\log (1-z) = \sum_{n=1}^{\infty}\frac{z^n}{n}$ for $z \in \mathbb{C}, \|z\|=1, z \neq 1$ ? In any case, why?
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0answers
33 views

Show that $\sum_{n=1}^{\infty} \frac{z^n}{n}$ converges for $z \in \mathbb{C}$ such that $\|z\|=1$ but $z \neq 1$

I know I could use Dirichlet's test, but I am wondering if the Taylor series of $- \ln (1-z)$ can be used in some way to prove it for $\|z\|=1$, $z \neq 1$. I know the convergence radius is 1 so it is ...
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4answers
45 views

Convergent complex series

Is $$\sum\limits_{n=1}^\infty \frac{i^n}{n} $$ convergent? Im confused as to how to solve this question, I've been trying to use ratio test but that doesn't seem to be helping.
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2answers
33 views

Integrated series identity with Legendre Polynomials

The Legendre Polynomials can be defined in many different ways and have several properties. Many of these can be found in books or in the net, but I couldn't find this one anywhere: Prove that: ...
0
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1answer
130 views

Difference between power series method and Frobenius method

There is the power series method for solving ordinary differential equations: one looks for solutions of the form $\sum c_n x^n$, and derives algebraic relations between coefficients $c_n$. Then ...
0
votes
2answers
42 views

Calculate the power series

I want to find the power series of $\frac{1}{3!}$ in the field $\mathbb{Q}_3$. In order to do this, do I have to solve the congruence $3!x \equiv 1 \pmod{3^n} \Rightarrow 6x \equiv 1 \pmod 3$? If ...
0
votes
1answer
31 views

Is this method of finding range of x for which given series is convergent, wrong?

There was this question in our midsem question paper: We had to find out the range of values of x (x is positive) for which the given series is convergent, Given series was $\sum_{n = 1}^{\infty} (a ...
5
votes
1answer
50 views

Can a power series converge uniformly on $(-1,1)$ but not on $[-1,1]?$

I am taking a course in analysis, and I am wondering whether it possible for a power series with radius of convergence $1$ to converge uniformly on $(-1,1)$ but not on $[-1,1]?$ I don't think this ...
0
votes
2answers
39 views

Prove that the limit of a series, containing 1/{powers-of-2}, is not rational

I have a series, $$x_n = \sum_{k=0}^n2^{-k^2-k}, \forall n \in N$$ I have to find it's limit and prove it is not in Q(it is not rational). I tried to write it $x_n=1+\frac{1}{2^1*2^1}+\frac{1}{2^4* ...
2
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0answers
32 views

How to compute the following sum?

How to compute the following sum? $$\sum_{k=1}^{\infty} \frac{k^{k-1} \cdot e^{-k}}{k!}$$ It is likely to be equal $1$ (there is an argumentation that goes back to random graphs). Moreover, i think ...
0
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0answers
16 views

Power series convergent at exactly one point of unit circle [duplicate]

I have to give an example of power series with radius of convergence equal 1 that is convergent at exactly one point of unit circle and divergent at all other points of that circle.
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0answers
50 views

Expanding in powers of $\epsilon$ and big O notation

I do not understand how to approach (D.1) equation Where did that big O notation come from?Is it using taylor series and linear approximation? Thanks in advance
5
votes
1answer
61 views

Differentiation under the integral sign and counting measure

Consider a power series $f(x)=\sum_{n=1}^\infty a_nx^n$, and assume that $\displaystyle R=\lim_{n\to \infty} \frac{a_n}{a_{n+1}}$ exists. Use differentiation under the integral sign to show that ...
3
votes
2answers
63 views

notation for first and second derivatives of a power series

I have a power series $$\sum_{k=0}^\infty\frac{c_k}{k!}x^k$$ where $c_k$ is an arbitrary $k$-th term of some sequence. Then ...
0
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0answers
8 views

Differentiational equation construct power series expansion

I got a question In order to improve the accuracy of your numerical estimate you are to use a power series expansion of y(x)to estimate y(1). (You may find it easier if you multiply both sides of ...
0
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1answer
29 views

Radius of convergence of $\sum a_n x^{n^2}$

Let $\sum a_n x^n$ be a power series with radius of convergence $R$. What is the radius of convergence of $\sum a_n x^{n^2}$? Can anyone help me here?
2
votes
2answers
26 views

Find the Power Series

How would one write $f(z) = \frac{1}{1-wz}$ as a power series? ( Where $z,w$ are in $C$.) Would it just be $\sum_{n=0}^{\infty} (zw)^n$?
2
votes
0answers
30 views

Taylor polynomial converging pointwise but not uniformly?

Many standard examples of Taylor series $(\exp(x), \sin(x), \cos(x))$ converge uniformly, others don't converge to its original function at all, e.g. $\exp(-x^{-2})$. I couldn't think of any smooth ...
1
vote
3answers
82 views

If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3$, what can be said about convergence at $x=2$ and $x=3$?

Problem: Is the following True or False: If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3 \implies:$ a) $\sum_{n=0}^\infty c_n 2^n$ converges. b) $\sum_{n=0}^\infty c_n 3^n$ converges. ...
3
votes
1answer
50 views

Writing solution to an arbitrary ODE with arbitrary initial values as the sum of a power series?

How can we solve for $y$ with these arbitrary initial values and polynomials? How would we write the solution as a power series?
0
votes
1answer
23 views

Radius of convergence of $2^n+3^n, n \geq 1$

Find the radius of convergence of the power series where, $a_n= 2^n+3^n, n \geq 1$. The answer is given to be 1. The tests I can use are Cauchy Hadamard Test and Ratio Test. My attempt: Using Ratio ...
0
votes
1answer
22 views

Radius of convergence of $1+3x+\frac{3^2x^2}{2!}+\cdots$

The question is to find the radius of convergence of the power series $1+3x+\frac{3^2x^2}{2!}+\frac{3^3x^3}{3!}+\cdots$ The answer is given to be $\frac{1}{3}$ My attempt: $a_n=\frac{3^n}{n!}$ ...
2
votes
0answers
15 views

Derivation of higher order bessel function in terms of lower order functions

I am really stuck trying to prove this.. ((x^-p)Jp(x))’ = -(x^-p)Jp+1(x) ---(1) Can someone please help how to actually prove this step by step, because whichever notes i see, they prove ...
0
votes
2answers
16 views

What is $1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$?

I want to express $$1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$$ in terms of standard functions (exp, cos, sin, etc.), but I just don't see what this function is. Does anybody here have an ...
1
vote
1answer
31 views

Recognising that $\sum_{n=0}^\infty \frac{a^2-b^2(2n+1)^2}{(a^2+b^2(2n+1)^2)^2}=-\frac{\pi^2\mathrm{sech}^2\left(\frac{a\pi}{2b}\right)}{8b^2}$

So I know from Mathematica that: $$\sum_{n=0}^\infty \frac{a^2-b^2(2n+1)^2}{(a^2+b^2(2n+1)^2)^2}=-\frac{\pi^2\mathrm{sech}^2\left(\frac{a\pi}{2b}\right)}{8b^2}$$ I am wondering how someone could ...
0
votes
1answer
28 views

Radius of convergence: $\sum_{k=1}^\infty \frac{x^{2k-1}}{2k-1}$

It is asked to find the radius of convergence of the series $$\sum_{k=1}^\infty \frac{x^{2k-1}}{2k-1}$$ i.e, to find the values of x such that this series converges. Clearly, I could directly apply ...
2
votes
1answer
31 views

Convergence of $\sum(-1)^k\frac{(\ln k)^p}{k^q}$ where $p,q$ in positive $\mathbb{R}$

For any $p, q$ in positive $\mathbb{R}$ $$\sum_{k=2}^{\infty}(-1)^k\frac{(\ln k)^p}{k^q}$$ I want to Use alternative series test for convergence but I'm struggling to verify that $\frac{(\ln ...
0
votes
0answers
31 views

statements of matrix analysis

Let $y$ be fixed value. Let $A=a(x,y)$ be a matrix and $f_{t}(x)=\frac{\sum_{n=0}^{\infty}{a^{(n)}(x,y)(\frac{1}{t})^n}}{\sum_{n=0}^{\infty}a^{(n)}(y,y)(\frac{1}{t})^n}$ Show that ...
5
votes
1answer
85 views

Twisted logarithm power series

I recently encountered a power series similar to the one of the $\log(1-x)$ of the form $$ F(x)= \sum_{n=1}^\infty \frac{\psi(n)x^n}{n}, $$ where $\psi$ is some Dirichlet character. Has anyone here ...
1
vote
1answer
71 views

Calculating the power series expansion about $\pi/2$ of $g(z)=\tan(z/2)$

Calculate the power series expansion about $\pi/2$ of $g(z)=\tan(z/2)$. Now calculate the expansion about $0$. I'm having trouble doing this. I'm not even sure which is the best way to approach ...
5
votes
0answers
87 views

Can a Power Series tell when to stop?

The naive description of the radius of convergence of a complex power series is as the largest radius so that the ball avoids poles and branch cuts. This makes sense in a world where analytic ...
1
vote
2answers
26 views

how to find power series in closed form

find a "closed form" of summation n=2 to infinity n(n-1)x^n. I don't have much clue to solve this can anyone please explain how to approach this?
2
votes
0answers
127 views

What more can i do to this infinite sum?

This question sprung out from another post of mine that was in part by Semiclassical, he Proved the Following: $$ \sum_{n=0}^{\infty} {}_2F_1(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1)/n! = 2\pi ...