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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5
votes
2answers
32 views

Deducing the series expansion of $\arctan(x^2)$ via the series expansion of $\arctan(x)$ at $x=0$

Comparing the series expansion of $\arctan(x^2)$ and $\arctan(x)$ at $x=0$ it looks like one can take the result from $\arctan(x)$ and replace each $x$ with $x^2$ to deduce the series expansion of ...
1
vote
0answers
7 views

About Convergence of a series

Is this series convergent? $$\sum_{n=0}^{N-1}\frac{c_{n}^{N}}{c_{N}^{N}n^2}$$ where $c_{n}^{N}$ is coefficient $x^{n}$ in chebyshev polynomial $T_{N}(x)$, i.e. ...
1
vote
1answer
44 views

Power series converging at the convergence radius

Let $f(z)=\sum a_nz^n$ be a power series of radius $R$. By Abel' radial theorem, if $f(R)$ converges then $f$ is continuous over real numbers at $R^-$. I had some questions on how that can be ...
0
votes
2answers
22 views

how to prove this statement related to radius of convergence

Suppose that the power series $$\sum b_nx^n$$ converges for $|x|$ less than or equal to $1$. Suppose that for some $s$ greater than $0$, $p(x)=0$ for all $|x|$ less than $s$. How to show that ...
1
vote
1answer
23 views

What is the radius of convergence of following series?

suppose that $\sum b_n$ is conditionally convergent but not absolutely convergent. What is the radius of convergence of the following power series $p(x)=\sum b_nx^n$?
0
votes
1answer
20 views

For what $x,y$ does $\sum_{k,l\ge 0} \frac{(k+l)!}{k!l!} \left| x^ky^l\right|$ converge?

For what $x,y$ does $\sum_{k,l\ge 0} \frac{(k+l)!}{k!l!} \left| x^ky^l\right|$ converge? I think that $\sum_{k,l\ge 0}\left| x^ky^l\right|$ will converge for $|x|<1$ and $|y|<1$ since ...
1
vote
1answer
22 views

Solving Laguerre coefficients with Integral?

I'm having some difficulty understanding the solution to a particular Laguerre expansion. The problem reads "Expand the term $ e^{-x}$ as a Laguerre expansion, noting the orthogonality of $$ < ...
0
votes
1answer
24 views

how to write as geometric series $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ [on hold]

How would I write $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ as a sum of geometric series?
2
votes
1answer
63 views

Find the radius of convergence and interval of convergence of the series

Find the radius of convergence and interval of convergence of the series: $\sum_{n=1}^{\infty}n^n x^{n^4}$ I'm really lost as to how to approach this problem. The other power-series problems were ...
1
vote
1answer
25 views

Function whose $n$-th derivative at $x=0$ is $n^3$ / evaluating the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$

I have troubles proving that the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$ represents the function $f(x)=e^x(x^3+3x^2+x)$. My idea was using the identity theorem for power series and the ...
2
votes
0answers
45 views

In what sense 1 + 0 + 2 + 0 + 3 + 0 + … = 1/24?

We know that the series $$ 1+2+3+\cdots=-\frac{1}{12} ~~ (1) $$ and $$ 0+1+0+2+0+3+\cdots=-\frac{1}{12} ~~ (2) $$ belong to the elementary Ramanujan class $R=4$ (definition, also here) and the series ...
1
vote
1answer
17 views

How does the speed of convergence of these formulae for calculating PI compare with the best algorithms?

I came across some series many years ago for calculating PI. I found that the first member of that series has been known for a long time in the math world. It is the set of series defined by: $$ ...
6
votes
0answers
71 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
0
votes
1answer
29 views

Show that entire function $f$ is a polynomial of degree at most $n$

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be a entire function. Suppose that there are $M$, $r>0$ and $n\in \mathbb{N}$ such that $\left|f(z)\right|<M\left|z\right|^n$ for all $z \in \mathbb{C}$ ...
0
votes
4answers
56 views

Show the given series is a solution of $y''-xy'-y=0$

My problem is this: "Show that the function represented by the power series, $$y=\sum_{n=0}^{\infty} \frac{x^{2n}}{2^nn!}=1+ \frac{x^2}{2}+ \frac{x^4}{8}+ \frac{x^6}{48}+...$$ is a solution of the ...
0
votes
0answers
25 views

Lang's proof of the Weierstrass preparation theorem

Relevant Google Books link. I'm having problems with the final step in the proof of Theorem 9.1. It's not clear to me why the function $I + \tau \circ \frac{\alpha(f)}{\tau(f)}$ should be ...
1
vote
0answers
29 views

In a recurrence relation, how do we know which order to terminate?

By employing Frobinious or Power Series approach, we my come up with a recurrence relation that is only solvable if we set any constant lower than $a_0$ or higher than $a_n$ vanish. For example, in ...
0
votes
0answers
31 views

Show that 1 + 2 + 0 + 4 + 0 + 0 + 0 + 8 + … = -1.

The diluted series of powers of $2$ $1+2+0+4+0+0+0+8+\cdots$ belongs to the elementary Ramanujan class $R=2$ and is summable to $-1$ (definition, also here). How to prove that result given ...
0
votes
2answers
26 views

If $a_k\ge 0$ for all $k$ show that $\sum\limits_{k=0}^na_k\le b\in\mathbb{R}$.

Given that $\lim\limits_{x\rightarrow1^-} \sum\limits_{k=0}^\infty a_kx^k = b \in\mathbb{R}$ for $|x|<1$. If $a_k\ge 0$ for all $k$ show that $\sum\limits_{k=0}^na_k\le b$. This is just a step in ...
0
votes
0answers
22 views

Neat expression for finite series with poisson distribution

I have the following expression $$ \sum_{n=1}^N f(k, n, p)\frac{1}{n} $$ where $f()$ is the binomial probability mass function: $$ f(k, n, p) = {n \choose k} p^k (1-p)^{n-k}$$ I wonder whether ...
0
votes
1answer
23 views

Converse of Abel's theorem

I know that a non conditional converse of Abel's theorem is not true, but is there a proof for the converse given certain conditions. So if $f(x)=\sum_{k=0}^\infty a_kx^k$ converges when $|x|<1$ ...
0
votes
2answers
16 views

Radius of convergence for complex power series

I am supposed to find the radius of convergence for the complex power series $$\sum_{n=0}^{\infty}(-1)^n2^nz^{2n+2}$$ I know that the radius of convergence is calculated by ...
-1
votes
0answers
27 views

Is the composition of an harmonic function with an analytic function an harmonic function in any dimension?

I was wondering if it is true or not that, given a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ and $g:\Omega\subset\mathbb{R}^n\rightarrow \mathbb{R}^n$ such that $f$ is harmonic and $g$ real ...
3
votes
2answers
38 views

True or false.If the series converges for x=1.1, then it converges for x=7

I saw this question in a previous year test and it seemed pretty simple, and that can often mean that I am missing something. If the series $$\sum_{n=0}^{\infty}a_n(x-3)^n$$ converges for $x=-1.1$, ...
1
vote
2answers
41 views

Limit of power series with L'Hospital

Calculate the given limit: $$\lim_{x\to 0} \frac{1}{1-\cos(x^2)}\sum_{n=4}^\infty\ n^5x^n$$ First, I used Taylor Expansion (near $x=0$): $$1-\cos(x^2)\approx 0.5x^4$$ I'm now quite stuck with the ...
4
votes
3answers
68 views

Find the sum of the $\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$

Let $0<p<1$,Find the sum $$\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$$
3
votes
2answers
58 views

What is the minimal correction to the harmonic series such that it converges?

as you all hopefully know, the series $$ \sum_{k\ge 1}\frac{1}{k} $$ diverges. Now I know that you can add some logarithmic corrections, such that it converges: $$ \sum_{k\ge 1}\frac{1}{k\log(k)^2} $$ ...
1
vote
0answers
41 views

Can you make a power series for $y=x^2$?

I tried to make a power series for $y=x^2$ by starting with $f^{-1}(x)=\sqrt{x}$ and applying Lagrange Inversion theorem with $a=1$, but it didn't converge. In fact, the best you could observe from ...
0
votes
0answers
22 views

What's the intuition behind this example of a power series converging everywhere on the boundary but not absolutely?

The example is $$\sum_{i=1}^\infty a_i z^i \text{ where } a_i = \frac{(-1)^{n-1}}{2^nn}\text{ for }n=\lfloor\log_2(i)\rfloor+1\text{, the unique integer with }2^{n-1}\le i < 2^n$$ It seems that ...
0
votes
1answer
20 views

How To Determine The Radius of This Power Series

$$ \sum_{n\ge 0} (3+\cos n)x^n ; a_n = (3+\cos n) $$ I used d'Alembert : $$\lim_{n\to\infty}\frac{a_{n+1}}{a_n} = \lim_{n\to\infty} \frac{3+\cos(n+1)}{3+\cos n} $$ Nw I'm stuck With How To get Rid ...
3
votes
1answer
21 views

Convergence radius and two-times-differentiability of power series.

I wanted to compute the radius of convergence for the following the power series $$\sum_{n=1}^{\infty} a_nz^n$$ with $(i) \, a_n = n!, \, (ii) \, a_n = \sqrt[\leftroot{-3}\uproot{3}n]{n}$ Then I ...
0
votes
0answers
22 views

$\sum\limits_{k=0}^{\infty}a_k(z+4)^k$ with $a_{2j}=(\sqrt{3})^{2j}$, different solutions

I want to calculate the radius of convergence of the series $$\sum\limits_{k=0}^{\infty}a_k(z+4)^k$$ where $a_{2j}=(\sqrt{3})^{2j}$ and $a_{2j-1}=\frac{1}{2j-1}$. I would calculate the radius of ...
1
vote
2answers
28 views

Simplifying Power Series as a Summation - Alternating Coefficients

I'm currently trying to rewrite a power series I have into summation notation. The series is as follows: $$ 2x + 3x^{4} + 2x^{7} + 3x^{10} + 2x^{13} + ... $$ Obviously I'll have $x^{3n+1}$ in the ...
1
vote
3answers
42 views

Obtaining the value of a power series similar to sine

I apologies for the vague title and the very specific question. I would like to know what $$K=4\left[ \frac{1}{1\cdot2!}-\frac{1}{3\cdot4!}+\frac{1}{5\cdot6!}-\cdots \right]$$ evaluates to. This is ...
1
vote
1answer
23 views

An elementary introduction to Puiseux series?

While studying Analytic combinatorics of Flajolet and Sedgewick (to be more specific, the coefficient asymptotics of algebraic functions), I have come across the concept of Newton-Puiseux expansions. ...
1
vote
4answers
48 views

How do I evaluate this series?

How do I evaluate this series: \begin{equation} \sum_{n=2}^\infty \frac{\prod_{k=1}^{n-1} (2k-1) }{2^nn!} = \frac{1}{8} + \frac{1}{16} + \frac{5}{128} + \frac{7}{256} +\ldots \end{equation} I ...
2
votes
1answer
25 views

Find the Taylor series and evaluate at $f^{39}(0)$

$$e^{-x^2}$$ I've had a hard time understanding power series since as long as I can remember. To my understanding, the question is asking me to write out the terms in the formula for Taylor series, ...
0
votes
1answer
28 views

Radius of convergence of the solutions of the differential equation

Justifies that the solutions are analytic functions in $t_0=0$ . Is it possible to determine the radius of cnvergencia series corresponding powers without calculate? $$ (1-t^2)x''-2tx'+a(a+1)x=0$$ ...
0
votes
1answer
43 views

What is known of convergence and divergence of the following series?

Let the serie $\sum_{k \geq 0} a_k (z-i)^k$ converge for $z = 4$ and diverge for $z=-8$. What is known of convergence and divergence of the following series? (a) $\sum_{k \geq 0} a_k (1+i)^k$ (b) ...
1
vote
3answers
30 views

Convert $f(x)=(\cos(x))^3$ to powers of x and find if converges.

I started out by writing the Taylor series for $x_0=0$ (Maclaurin series) of $f(x)=(\cos(x))^3$. If my calculations are correct $$f(x)=1-\frac{3x^2}{2!}+\frac{21x^4}{4!}-\frac{183x^2}{6!}+...$$ and ...
1
vote
0answers
53 views

Differential equation: $t^2x''+tx'+(t^2-3)x=0$

We have the following differential equation: $$ t^2x''+tx'+(t^2-3)x=0$$ Give the set of solutions of the differential equation What solution $x=x(t)$ check that $$ \lim_{t\rightarrow ...
1
vote
2answers
84 views

Over an integral arising from Kepler's problem [also: generally useful integral, NOT DUPLICATE!]

This post might appear as a duplicate of the following: Over an integral arising from Kepler's problem [also: generally useful integral] So recalling quickly: $$\Phi(\epsilon) = ...
1
vote
1answer
31 views

Solve differential equation centering in 1

Hi I try solve the following problem of differential equation $$ x''+tx'+\frac{1}{1+t+t^2}x=0 $$ I have to solve that differential equation using power series centering in 1, but I do not know how ...
0
votes
3answers
58 views

Series expansion: $1/(1-x)^n$

What is the expansion for $(1-x)^{-n}$? Could find only the expansion upto the power of $-3$. Is there some general formula?
0
votes
3answers
55 views

Prove that $\sum_{n=1}^\infty \frac{x^n}{n(n+1)}$ converges [closed]

Prove that the following power series converges: $$\sum_{n=1}^\infty \frac{x^n}{n(n+1)}$$ I have tried using d'Alembert's ratio test however this was inconclusive. Anyone have any ideas?
0
votes
1answer
27 views

Power series confusion when multiplying fractions.

I am stuck on the following question. check that the following sum from 0 to infinity converges using power series. sum of $$ 1/((n+(1/2))^2)$$ the next line of work is : $$4/((2n+1)^2)$$ I have ...
-1
votes
1answer
27 views

Prove that ${}_2F_1(0,b;c;z)=1$

I do not know how I could prove that ${}_2F_1(0,\beta;\gamma;t)=1$ because when I apply the definition I get $0$, namely.. $$ \sum_{n=0}^{\infty}\frac{(0)_n(\beta)_n}{n!(\gamma)_n}t^n=0$$ someone ...
3
votes
0answers
42 views

Problem: differential equation

Hi I try solve the following problem of differential equation $$ x''+tx'+\frac{1}{1+t+t^2}x=0\tag 1$$ when $$x(1)=0\ \ \ ;\ \ \ x'(1)=1 $$ is the solution analytic in $t_0=1$ and his convergence ...
3
votes
0answers
43 views

Can all series in the elementary Ramanujan class R = 2 be shifted?

For $f(x)=\sum_{n=0}^\infty a_nx^n$ and $g(x)=f(x)-Rf(x^2)$, $R\neq1$, $f(1)$ belongs to the elementary Ramanujan class $R$ if $g(1)$ is Abel summable. The elementary Ramanujan sum of $f(1)$ is ...
1
vote
2answers
71 views

Looking for “an easy to understand” proof for following Power series

I'm looking for proof for the following Power series $exp(X) = \sum_{k=0}^{n} \frac{X^{k}}{k!}$ If X is $A_{nxn}$ matrix, then prove the series is converge Given $\exp(X) = \sum_{k=0}^{n} ...