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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16 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 ...
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0answers
30 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 ...
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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 ...
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0answers
21 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 ...
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1answer
20 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 $|x|<1$ ...
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2answers
14 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 ...
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24 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 ...
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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 $x=-1.1$, ...
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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 ...
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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 $$\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$$
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2answers
57 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: $$ \sum_{k\ge 1}\frac{1}{k\log(k)^2} $$ ...
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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 ...
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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 ...
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1answer
19 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 ...
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1answer
19 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 ...
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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 ...
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2answers
27 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 ...
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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 ...
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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. ...
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4answers
47 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 ...
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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, ...
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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? $$ (1-t^2)x''-2tx'+a(a+1)x=0$$ ...
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1answer
41 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) ...
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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 ...
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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 ...
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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) = ...
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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 ...
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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?
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3answers
53 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?
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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 ...
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1answer
26 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
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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 ...
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0answers
40 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 ...
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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} ...
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4answers
33 views

power series for $(a+x)^{-1}$

Is it possible to write the following expression in terms of power series? $ (a+x)^{-1}=\sum\limits_{k = - \infty }^\infty {{b_k}{x^k}}$ where $0<a<1$ and$0<x<1$.
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2answers
37 views

How to prove that this sequence is divergent? Problem with a limit

I'm trying to calculate the interval of convergence and the values of x in which the series converge absolutely and conditionally $\sum _{n=1}^{\infty }\:\left(1+\frac{1}{\:n}\right)^nx^n$ My ...
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0answers
29 views

Is it possible to very roughly approximate a Taylor Series expansion? [closed]

Say I have the following equation: $$ y=-0.000001x^6+0.00001x^5+0.0001x^4+0.001x^3-0.01x^2-0.1x+1 $$ Is it ever possible to find a Taylor polynomial that is similar enough (but not exactly the ...
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4answers
31 views

Find power series using long division

How do you find the power series of $\frac{1}{2+x}$ using long division? When I do the long division, I get zero with a remainder of 1, giving me the original function back.
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1answer
20 views

Determining if $A(B(x))$ is a formal power series

I know that by theorem, $A(B(x))$ is a formal power series if $b_0=0$. I wasn't sure if it works the other way around. Can I also say that if $b_0\ne0$, $A(B(x))$ is not a formal power series? If it ...
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0answers
14 views

Solving $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $

I would like to work out the result of $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $. Here, $t, i, N_i, m_i$ are positive integers. My effort: $$ \Pi^t_i 2 m_i \left(N_i!\right)^{m_i} \implies (2 m_1 ...
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2answers
40 views

Coefficient of a term in a formal power series

I just started learning about formal power series and I'm trying to find the coefficients in the expansion of a formal power series, specifically $$[x^6](1-3x^2)^{-9}$$ Using the binomial theorem and ...
2
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2answers
48 views

MacLaurin series for $9\sec(3x)$

A question I've been given asks me to find the first 3 non-zero terms of the MacLaurin series for the function: $y = 9sec(3x)$ Looking at old questions on this forum, I think that this is supposed to ...
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0answers
14 views

Uniqueness set for analytic functions of several variables

Is there a simple (and not so restrictive) condition for a set to be an uniqueness set for the space of holomorphic functions defined on some open subset $U \subseteq \mathbb{C}^n$? By uniqueness set ...
3
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2answers
58 views

Maclaurin Series of $e^{-x^2}$

The question is: Find the first 3 non-zero terms in the MacLaurin series for the function: $$y =e^{-x^2}$$ I have been told to simply substitute the $-x^2$ into the standard MacLaurin series for ...
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1answer
43 views

To find Radius of convergence of series $1 + \frac{1.x^2}{2.3} + \frac{1.3.x^4}{2.4.5} + \frac{1.3.5.x^6}{2.4.6.7} $

The radius of convergence of series $$1 + \frac{1.x^2}{2.3} + \frac{1.3.x^4}{2.4.5} + \frac{1.3.5.x^6}{2.4.6.7} $$ I apply ratio test and got R.O.C = 1. But textbook states answer to be 1/2.I don't ...
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2answers
57 views

Infinite series

Is it possible to compute explicitly or in an approximate way the infinite series: $$ S(x) = \sum_{k=0}^{+\infty}{\frac{x^k}{k!} (a+k)^{-(n+1)}}$$ where $n \in \mathbb{N}$.
1
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1answer
29 views

Looking to have my worked check on a calculus series question

I am trying to determine if the Taylor series of $f(y) = y^{-\frac{1}{3}}$ about $y=1$ converges absolutely at $y = 2$. I am calculating the Taylor series as $$f(y) = 1 + a_1 (y-1) + a_2 (y-1)^2 + ...
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1answer
22 views

Two convergent power series are the same if they equal on an infinite set of points having 0 as a limit point.

I'm having difficulty following the proof of the theorem below. First of all, how do we know that h(z) is a power series having a non-zero radius of convergence from the fact that f(z) is. And, ...
0
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1answer
12 views

Factoring to find Power Series and Radius of Convergence

The question asks: Find a Power Series representation and Radius of Convergence for: $f(x)=\dfrac{x}{9+x^2}$ I see that it is pretty straight forward that if I re-write this by factoring out a 9, ...
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0answers
22 views

Convolution vs term wise product

Fix an element $x\in\mathbb R^\times$. Then I can define the ring $\mathbb R[[x]]$ which is the set: $$\left\{\sum_{i=0}^\infty a_ix^i\,: a_i\in\mathbb R\right\}$$ and where the product is the usual ...