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

0
votes
0answers
20 views

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

Let $f(t), g(t)$ be polynomials, and let $y$ be a function of $t$. Given the ODE $y'' + f(t) y' + g(t) y = 0$ with initial conditions $y(0) = \alpha$ and $y'(0) = \beta$, write $y$ as the sum of a ...
0
votes
1answer
19 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
11 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
15 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 ...
0
votes
0answers
65 views

Why is such a the series algebraic but rational?

The coefficients of the series expansion of the algebraic function $A=\frac{1-\sqrt{1-8x^2}}{4x}$ are all intergers: $$A(x)=x+2x^3+8x^5+\cdots$$ But according to Polya's research,if $ F(x)$ is a ...
1
vote
1answer
26 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
25 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
24 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
30 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 ...
0
votes
0answers
9 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 ...
0
votes
0answers
27 views

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

Calculating 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 it, for ...
5
votes
0answers
72 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
23 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?
1
vote
0answers
39 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 ...
0
votes
1answer
20 views

asymptotic series for “stable distribution”

I'm trying to understand how to get from one equation to another in a certain paper I am studying (DOI:10.1080/00018738100101467, eqs. 4.34 and 4.35). The equations are pretty self contained, so I'm ...
0
votes
1answer
19 views

Prove the equality with power series

I have to prove for $|x| < 1$ that $$ \ln\frac{2(1-\sqrt{1-x})}{x} = \frac 12 \cdot \frac x2 + \frac 12 \cdot \frac 34 \cdot \frac{x^2}{4} + \frac 12 \cdot \frac 34 \cdot \frac 56 \cdot ...
2
votes
0answers
28 views

Help understand part of the proof. Radius of convergence is $\frac{1}{\limsup |a_n|^{1/n}}$

Can you help me understand the highlighted parts of the proof. Thanks :) Theorem: Let $\sum{a_nz^n}$ be a power series, let r be its radius of convergence. Then $\frac{1}{r} = \limsup |a_n|^{1/n}$. ...
-2
votes
0answers
25 views

show that very positive integer n can be expressed uniquely in the form 2^t+k(t)2^(t+1) [closed]

Show that very positive integer n can be expressed uniquely in the form 2^t+k(t)2^(t+1) Than use this fact to show that for |z| less than 1 we have ...
0
votes
0answers
16 views

Explain $z(\cos z -1)$ serie expansion

Look at the following expansion, which should be an expansion from for the coefficients $a_0, a_1, a_2, a_3$ $$\begin{align} z(\cos z -1) &= z \left( 1 - z^2/2!+ z^4/4! - z^6/6! + ...
1
vote
0answers
29 views

Ideal in power series ring

Let $J$ be an ideal in $k[[x_1,...,x_n]]$ such that $(x_{1},...,x_{n})^{2}\subseteq J$, $\{x_{1},...,x_{r}\}\nsubseteq J$ and $\{x_{r+1},...,x_{n}\} \subseteq J$, for some $1\leq r \leq n$. I want to ...
0
votes
2answers
33 views

Cannot expand $\sin(2x^2-4x+3)$ at $x_0 = 1$

Trying to expand $\sin(2x^2 - 4x+3)$ at $x_0 = 1$ to the $O(x-x_0)^n$. After substitution $t = x - 1 $, the problem becames $$\sin(2t^2+1) \text{ at } t_0 = 0$$ While we know that $$\sin(s) = ...
1
vote
2answers
43 views

Estimating the behavior for large $n$

I want to find how these coefficients increase/decrease as $n$ increases: $$ C_n = \frac{1}{n!} \left[(n+\alpha)^{n-\alpha-\frac{1}{2}}\right]$$ with $\alpha=\frac{1}{br-1}$ and $0\leq b,r \leq 1$. ...
1
vote
0answers
8 views

Simplifing a Cauchy product to find the recurrence relation when solving a differential equation using a power series solution.

I'm having trouble finding the recurrence relation of the following non linear differential equation: $y''(x)+p(x)y'(x)+y^2(x)=0$ with $y(0)=1$ and $y'(0)=0$ where ...
1
vote
1answer
21 views

Vanishing of Taylor series coefficient

I am solving previous year question paper some competitive exam. Give me some hint to solve the following problem. Let $f$ be an entire function. Suppose for each $a \in \mathbb{R} $ there exists at ...
1
vote
2answers
24 views

Importance of the first term in a Taylor series

Suppose you have a function $f(x)$ whose Taylor series can be represented as the power series $$a_0 + a_1x^2+a_2x^4+...$$ If you are told that for $x\in\mathbb{R}_+$, $$a_0 + a_1x^2 + a_2x^4 + ...
1
vote
3answers
30 views

Compute the following sum for any x?

Compute the following sum for any x? $\sum_{n=0}^\infty {(x-1)^n\over (n+2)!}$ I am having trouble to compute that sum. It looks like geometric series but I don't know where to start. Can everyone ...
1
vote
1answer
24 views

Finding a Taylor Series representation of $f(x)=\ln(\frac{1+2x}{1-2x})$ centered at $0$.

I'm trying to find a Taylor Series representation of $f(x)=\ln(\frac{1+2x}{1-2x})$ centered at $0$. So I am using the Maclaurin Series representation of $f(x)=\ln(1+x)$ which is ...
2
votes
3answers
105 views

Estimating the series: $\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$

Any idea on how to estimate the following series: $$\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$$ where $a$ and $b$ are constant values. Greatly appreciate any respond.
0
votes
0answers
20 views

Expression for a series of squared sines

Does anyone know if there is a single expression for $$-\frac{1}{2}\sum_{j=1}^{\infty}\sin^2\left(\frac{2\pi x}{3^j}\right)$$ or at least a nicely-expressed upper bound? I've already computed that ...
1
vote
4answers
105 views

Power series in $\mathbb{Q}_5$

Could you help me to find the first five positions of the power series in $\mathbb{Q}_5$ of $\frac{1}{2}$? How can I do this? Is there a general formula?
0
votes
1answer
11 views

Use the power series representations of functions to find the taylor series of $\frac{1}{5+x'}$ at center = -6.

I am trying to find the taylor series of $f(x)=$ $\frac{1}{5+x'}$. And I cannot seem to get how to find the taylor series using the method I've been using for other functions. Another thing that's ...
5
votes
0answers
60 views

Is $\int f=f-1\iff f(\cdot)=e^{\cdot}$ proved this way correct?

I saw this on math overflow and made me wonder, why does it work, is it rigorous, can we really factor like this, and where can we use similar tricks; Let $\int$ denote $\int_0^x$ Then solve $$\int ...
1
vote
1answer
34 views

How to obtain this geometric progression

How do I obtain this from the formula of the geometric progression (which I 'only' know as $1+q+q^2+...+q^{n-1} = \frac{1-q^n}{1-q}$)? $$\frac{x_1^p-x^p}{x_1^q-x^q} = ...
0
votes
1answer
57 views

How can we show that it is an integer 5-adic number?

Show that the number $\frac{3}{8}$ is an integer $5$-adic and calculate the first five positions of its power series in $\mathbb{Q}_5$. Could you explain me how we can conclude that $\frac{3}{8}$ is ...
0
votes
1answer
22 views

differential equation using series expansion

Trying to solve xy'= xy + y using the series y(x) = $\sum\limits_{i=0}^\infty a_nx^n$ This is what i have so far. y'(x)= $\sum\limits_{i=0}^\infty na_nx^{n-1}$ xy' - xy - y = 0 x ...
0
votes
1answer
16 views

Finding the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{n!x^{n+9}}{(2n)!}$.

I am having trouble understanding how to find the interval of convergence/radius. I know the interval of convergence is $(-\infty,\infty)$, and the radius is $\infty$ of $\sum_{n=0}^{\infty} ...
4
votes
2answers
58 views

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

Find the sum $$\sum_{n=1}^\infty \frac{n}{(1+x)^{2n+1}}.$$ Indicating the interval of convergence for $x$. My attempt: Let $ t=\frac{1}{x+1}$. Then, applying the root test, $$\lim_{n\to \infty} ...
0
votes
2answers
43 views

What is the general formula for power series summation? [duplicate]

While reviewing definite integrals, $\int_a^bf(x)dx$; I recalled that a definite integral could not only be solved by the difference of the anti-derivatives of intervals b and a, $F(b)-F(a)$, via the ...
0
votes
0answers
17 views

Find at least the first four nonzero terms in a power series expension of the general solution about xo= 0

$x^2y''-y'+y = 0$ about $x_0 = 2$ I tried to get an answer by letting $t = x-2$ and assuming $$y = \sum_{n=0}^\infty a_nt^n$$ $$y' = \sum_{n=1}^\infty na_nt^{n-1}$$ $$y'' = \sum_{n=2}^\infty ...
4
votes
0answers
63 views

Generating Function for Associated Stirling Numbers $b(n,k)$

I am trying to identify or find the ordinary generating function (not the exponential generating function) for the Associated Stirling numbers of the Second kind, denoted $$b(1;n,k)=b(n,k)$$ These ...
6
votes
6answers
223 views

Prove that $\sum_{n=1}^\infty \frac{n^2(n-1)}{2^n} = 20$

This sum $\displaystyle \sum_{n=1}^\infty \frac{n^2(n-1)}{2^n} $showed up as I was computing the expected value of a random variable. My calculator tells me that $\,\,\displaystyle ...
1
vote
2answers
70 views

Exact value for a series

I would like prove this equality $$1-\frac{1}{n-1}+\frac{1}{n+1}-\frac{1}{2n-1}+\frac{1}{2n+1}-\frac{1}{3n-1}+\frac{1}{3n+1}+... =\frac{\pi}{n\tan{\frac{\pi}{n}}}$$
0
votes
1answer
36 views

Questions regarding a complex-analytic function

So the question is formulated as follows. Given the analytic function $z \mapsto f(z) = \dfrac{1}{\sin z} - \dfrac{\cos z}{z}$, Is $z = 0$ a pole, an essential singularity, a removable singularity, ...
3
votes
4answers
64 views

First four terms of the power series of $f(z) = \frac{z}{e^z-1}$?

Attempt: $$ e^z = \sum_{n=0}^\infty \frac{z^n}{n!}$$ $$ e^z - 1 = \sum_{n=0}^\infty \frac{z^n}{n!} -1$$ $$ e^z - 1 = z\sum_{n=0}^\infty \frac{z^n}{(n+1)!} $$ Thus $$ \frac{z}{e^z-1} = ...
0
votes
2answers
63 views

How does this converge uniformly?

My teaching assistant told me that $\sum \frac{(-1)^n x^n}{n}$ converges uniformly on $[0,1]$, but I doubt that. I can only see that it is uniformly convergent on $(-a,a)$ where $0<a<1$. How ...
0
votes
0answers
36 views

Power series and the average of its coeffients

Assume $\{b_t\}_{t\geq 0}$ is a bounded real sequence. Let $\{T_k\}_{k\geq 1}$ be an increasing sequence of natural numbers such that $\lim_k T_k=+\infty$. Suppose $$\lim_{k\rightarrow ...
1
vote
2answers
146 views

Regarding the radius of convergence and its equality to a certain limit

Let $f$ be a holomorphic function on the open unit disk $\mathbb{D}$, and suppose that $f$ cannot be extended holomorphically to any open set $\Omega$ containing $\overline{\mathbb{D}}$. Let $f(z) = ...
2
votes
1answer
94 views

Evaluate $ \int_0^1 \sum_{k=0}^\infty (-x^4)^k dx = \int_0^1 \frac{dx}{1+x^4} $

I have read this thread and I found in some comments the above named equality. I couldn't follow the transformation, which are done to get from the left to the right side at that point in particular. ...
1
vote
2answers
28 views

Find relative radius of convergence for this seies

Given two series $\sum _{n=1}^{ \infty} a_nz^n$ and $\sum _{n=1}^{ \infty} b_nz^n$ who both have radius of convergence $R$, show that the radius of convergence for $\sum _{n=1}^{ \infty} c_nz^n$ is at ...