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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3answers
124 views

Questions on the differential equation $df/dx=-[f(x)]^2$

I have another group project problem I am having trouble with. Here is the first part of the problem: "Consider the differential equation $df/dx=-[f(x)]^2$, with initial condition $f(0)=a$." ...
2
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1answer
18 views

General Case of Convergence of a Power Series

The Question If $f(x) = \sum c_nx^n$, where $c_{n+4} = c_n$ for all $n\ge 0$, find the interval of convergence of the series and a formula for $f(x)$ My Work and Question I haven't been able to do ...
1
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0answers
42 views

Calculating the Laurent Series of $\tan z$

I need help calculating the laurent series of $\tan z$ around the points $z=0$, $z=\pi/2$, and $z=\pi$. How would one go about doing this? I solved an almost identical question that was "Derive the ...
0
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1answer
22 views

If $S_n\to \infty$ as $n\to \infty$ is the following inequality valid or when is it valid? $\frac{a_n}{S_{n-1}}\leq \frac{C}{n}.$

Let $a_n$ be a sequences of positive real numbers and $S_n=\sum_{k=1}^{n}a_k.$ If $S_n\to \infty$ as $n\to \infty$ is the following inequality valid or when is it valid?$(C>0)$ ...
4
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3answers
46 views

Find the interval of convergence of $x + \frac{1}{2} x^2 + 3x^3 + \frac{1}{4}x^4 +…$

How to find the interval of convergence of the following series: $x + \frac{1}{2} x^2 + 3x^3 + \frac{1}{4}x^4 +...$ I have no idea what to proceed. Any help? Thanks!
2
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2answers
39 views

Calculate Laurent Series for $\frac{\ln z}{(z-1)^3}$ about $z=1$

Calculate the Laurent series of the function $g(z)= \frac{\ln z}{(z-1)^3}$ about the point $z=1$. Well since the singularity and the centre of the circle we are expanding about collide, I can just ...
2
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1answer
28 views

Finding the interval of convergence of $\sum_{n=2}^\infty \frac{x^n}{(\ln (n))^2}$

I have to find the interval of convergence of the following power series: $$\sum_{n=2}^\infty \frac{x^n}{(\ln (n))^2}$$ My approach to the problem: I start by using the ratio test: $$\lim_{n \to ...
2
votes
1answer
31 views

limit of $a_n(x)=\frac{\sum_{k=0}^{t_n}x^k}{\sum_{k=n+1}^{t_n}x^k}$

Let $t>1$ and $[t_n]$ defines the integer part of the product $t.n$. I need to evaluate the sequence $$a_n(x)=\frac{\sum_{k=0}^{[t_n]}x^k}{\sum_{k=n+1}^{[t_n]}x^k}$$ is convergent or bounded as ...
0
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1answer
101 views

Using power series to solve non-homogeneous differential equation?

I've been stuck on this for a while. I've got the following non-homogeneous differential equation and I have to give a solution in the form $\sum_{n=0}^{\infty}c_nX^n$: $$ y'' - 2y'x^2 + 4xy = x^2 + ...
0
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1answer
36 views

Finding when a series terminates

A recurrence relation $$\frac{b_{j+2}}{b_{j}} = \frac{j-\xi}{(j+2)(j+1)}$$ defines the general term for the power series where $\xi$ is a constant. $$g(x)=\sum^{\infty}_{j=0} b_{j}x^{j}.$$ I want to ...
10
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4answers
320 views

How does one show sin(x) is bounded using the power series?

Define the real valued function $$ \sin:\mathbb{R} \rightarrow \mathbb{R}, \qquad given ~~by \qquad \sin(x) := x-\frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots $$ How does one show ...
1
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1answer
46 views

Assuming the expansion for $(1+x)^{-1}$ prove that $\int_0^1 \frac{x \mathrm d x}{1+x}=\frac12-\frac13+\frac14-\frac15+\cdots$

Given: $(1+x)^{-1}=1-x+x^2-x^3+\cdots$ for $-1<x<1$, prove that $\int_0^1 \frac{x \mathrm d x}{1+x}=\frac12-\frac13+\frac14-\frac15+\cdots$ My attempt: I multiplied both sides of ...
0
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1answer
49 views

Prove that $ \frac12 [log(1-x)]^2 = \frac12x^2 + (1+ \frac12) \frac13 x^3+\cdots$ for $-1<x<1$

$\frac12 [log(1-x)]^2 = \frac12 x^2 + (1+ \frac12) \frac13 x^3+ (1+ \frac12 + \frac13) \frac14 x^4+ \cdots$ My attempt: I'm thinking of finding a series which is convergent in $-1<x<1$ and ...
6
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3answers
98 views

what's the summation of this finite sequence?

$a$ and $b$ are positive integers. The summation is $$\sum\limits_{x = 1}^a {x\left( {\begin{array}{*{20}{c}} {a + b - x}\\ b \end{array}} \right)} .$$ Any closed-form expression? I thought it ...
2
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2answers
78 views

Need explain how to find sum of series

Can someone explain me how to find sum of next series: $\sum_{n=1}^\infty n^4 \tan^{n-1}(x)$ Thanks for answers in advance.
0
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2answers
84 views

Bernoulli Numbers and radius of convergence

consider the function $f(x)=\frac{x}{e^x-1}$. Since the function $\frac{1}{f(x)}=\frac{e^x-^1}{x}=\sum\limits_{k=0}^{\infty} \frac{x^k}{(k+1)!}$ has a taylor expansion with $\frac{1}{f(0)}\neq 0$ we ...
0
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1answer
216 views

By applying term-wise differentiation and integration find the sum of the series $\sum_{k=1}^{\infty}\frac{x^k}{k}$

I need to find the sum of the following series: $$\sum_{k=1}^{\infty}\frac{x^k}{k}$$ on the interval $x\in[a,b], -1<a<0<b<1$ using term-wise differentiation and integration. Can anyone ...
2
votes
1answer
47 views

Product series general formula?

For a series like : $$1^3 + 2^3 + 3^3 + 4^3 + \cdots + n^3$$ There is a general formula : $(n(n+1)/2)^2$ My question: Is there any general formula possible for following series : $$1^1 \cdot 2^2 ...
2
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1answer
35 views

Numerical Evaluation of a Series at a Point

I have a numerical calculus challenge to resolve using a C++ algorithm or scilab. The problem is the following: $f(x)=\sum_{n=1}^{\infty }a_{n}x^{n}$ where $a_{n}=\sqrt{n^2+1}-n$ This function is ...
1
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1answer
26 views

Convergence of Power Series $\sum_{n=0}^{\infty}\frac{1+\alpha^{n}}{1+\beta^{n}}z^{n}$ with $\alpha ,\beta \ge0$

For $\alpha, \beta \geq 0 \in \mathbb{R}$, find the radius of convergence for the series: $$\sum_{n=0}^{\infty}\frac{1+\alpha^{n}}{1+\beta^{n}}z^{n}$$ Ok, so if $\alpha$ and $\beta$ are $\leq ...
0
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3answers
35 views

Finding the sum of x of two power series.

Could someone give me a hint on finding the sum of all $x$ for the following power series: $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{x^{2n+1}}{2n+1} $$ I am pretty sure we need to compare this with ...
4
votes
1answer
37 views

Evaluation of formal series

Is it possible to get a closed form for coefficients of $$\left(1+\frac{2t}{(1-t)^2}\right)^{-n}$$ there $n$ - positive integer? It's easy to obtain the formula for $m$-th coefficient as ...
1
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2answers
37 views

Method for solving radius of convergence problem

Hi I am interested if the following method for solving for the radius of convergence for power series problem is a valid method: Find the radius of convergence of the following: ...
0
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2answers
30 views

Method of finding radius of convergence

Hi is it acceptable to evaluate the radius of convergence $R$ of this power series $$\sum_{n=1}^{\infty}(-1)^{n}n^{-\frac{2}{3}}x^{n}$$ by instead of taking $a_{n} := (-1)^{n}n^{-\frac{2}{3}}$ we take ...
4
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1answer
32 views

prove this equality

First, I try to use the Taylor series but it doesn't work well. And someone said I can use fundamental theorem of calculus but I don't see it. can anyone give me a good hit? thanks
0
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2answers
14 views

How to prove this equality about series?

I'm trying to prove this equality (unsuccessfully): $$\sum_{n=0}^\infty \frac{i^nt^n}{n!}=\sum_{n=0}^\infty \frac{(-1)^nt^{2n}}{(2n)!}+i\sum_{n=0}^{\infty}\frac{(-1)^nt^{2n+1}}{(2n+1)!}\ , $$ which ...
2
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2answers
42 views

Radius of convergence of $\sum_{n=0}^\infty \frac{a_n}{c^n} x^n$ where $c\in\Bbb Z$ is a non-zero constant

So I have a power series $\sum_{n=0}^\infty a_nx^n$ with radius of convergence $R=1$. I then want to find the radius of convergence of $\sum_{n=0}^\infty \frac{a_n}{c^n} x^n$ where $c$ is a non-zero ...
0
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3answers
29 views

Taylor Series Expansion for 1/(1+z^2) [duplicate]

Given $f(z) = \frac{1}{1+z^2}$, I want to find the Taylor series of $f(z)$ about $z_0 = 0$. Intuitively, and based on the formation of a standard power series I have $f(z) = \sum (-1)^n(z^2)^n$. ...
0
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0answers
27 views

Power Series of the Principal Branch of Logarithm

How would I go about determining the power series expansion for the principal branch of $\log$ about $z = i$? I would assume I should start from $\frac{1}{z}$, but I don't know how to manipulate this ...
0
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1answer
52 views

Derive an explicit formula for a power series

Could anyone help me find an explicit formula for: $$ \sum_{n=1}^\infty n^2x^n $$ We're supposed to use: $$\sum_{n=1}^\infty nx^n = \frac{x}{(1-x)^2} \qquad |x| <1 $$
0
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0answers
21 views

Complex Power Series Convergence and Expansion

Can we have a power series of the form $\sum_{k=0}^\infty c_k(z-1)^k$ which converges at $z = 3$ but diverges at $z = 0$? This is for homework and I just don't know where to start. Tips would be ...
7
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2answers
136 views

Proof $1 -\frac{1}{5} + \frac{1}{9} - \frac{1}{13} + … = \frac{\pi + 2\ln(1+\sqrt2)}{4\sqrt2}$

I'm trying to show that $$1 -\frac{1}{5} + \frac{1}{9} - \frac{1}{13} + \cdots = \frac{\pi + 2\ln(1+\sqrt2)}{4\sqrt2}.$$ I thought of using the power series for $\tanh^{-1}z$ which I found was ...
0
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1answer
34 views

Series for $(1-z)^{-\frac{1}{2}}$ and application?

How could I obtain $$\sum_{k=0}^{\infty} {4k \choose 2k} \frac{z^{2k}}{2^{4k}}$$ from $$\sum_{k=0}^{\infty} {2k \choose k} \frac{z^k}{2^{2k}}$$ which is $(1-z)^{-\frac{1}{2}}$. I can't manage to ...
2
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4answers
50 views

What does it mean intuitively for a Taylor Series to be centered at a specific point?

I understand what a Taylor series is and how to find the Taylor series of a function. However I do not understand intuitively what it means to find a Taylor series for a specific function, centered at ...
1
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0answers
23 views

Product of two summations

Let $A$ be an open subset of $\mathbb{R}^n$. For an infinitely differentiable function $f(x)$ on A$, define $$[f](\rho) = \sum_{ \alpha \in \mathbb{N}^n} { \frac{\sup_{x \in A} | \partial_x^{\alpha} ...
2
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1answer
62 views

Integration rules for ∫ x^x dx [duplicate]

I'm working on Project Euler, problem 48: The series, 11 + 22 + 33 + ... + 1010 = 10405071317. Find the last ten digits of the series, 11 + 22 + 33 + ... + 10001000. This would be the ...
6
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1answer
45 views

Finding a bound for $\sum_{n=k}^l \frac{z^n}{n}$

For $z\in\mathbb{C}$ such that $|z|=1$ but $z\neq1$ and $0<k<l$, I'm trying to prove that: $$\left|\sum_{n=k}^l \frac{z^n}{n}\right| \leq \frac{4}{k|1-z|}$$ It's more of a game that slowly ...
1
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1answer
35 views

Expanding $(x+yi)^c$ to series

I need to evaluate a complex expression $f(x,y)=(x+yi)^c$, where $x,y,c\in\mathbb{R}$, in double-precision arithmetic on the GPU. It is done in a usual way, i.e., computing $\exp(c \log(x + yi))$. ...
2
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0answers
27 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
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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
votes
1answer
44 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
votes
2answers
42 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
vote
2answers
29 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
108 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
125 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
51 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 ...