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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3
votes
2answers
75 views

$\frac{1}{x^2} \int xe^x dx$ without using integration by parts

On a test i just had, i needed to solve a differential equation which lead me to having to find the result of $$ \frac{1}{x^2}\int xe^x dx $$ I then attempted to do this integral without integration ...
1
vote
1answer
26 views

How to prove matrix geometric convergence to any matrix?

Suppose I have two vectors $x$ and $v$, and we want to calculate the following expression: $$(I+x\cdot v^{T})^{-1}$$ My professor affirmed that we could treat this as a "geometric progression" ...
1
vote
2answers
147 views

Finding Sum for Infinite Series

Normally when I keep try multiple ways to solve a problem, I get an idea of where to start, and eventually can solve it. But it hasn't been the way for this question and I've been stuck for hours. ...
1
vote
1answer
31 views

why are these two power series the same

$$-\sum_{\color{red}{n=1}}^{\infty}nc_{n}x^{n}=-\sum_{\color{red}{n=0}}^{\infty}nc_{n}x^{n}$$ How come one starts at $1$ and the other starts at $0$ yet their equal? Do they both equal infinity?
1
vote
1answer
26 views

Where the power series is convergent

Where $f(z)=\sum_{n=1}^{\infty}\frac{(2i)^n}{n}z^n$ is convergent? I checked that the radius of convergence is equal to $\frac{1}{2}$. Now, since we know that the series ...
0
votes
1answer
23 views

Laurent Series Expansion for $f(z)=\dfrac{z+2}{(z+1)(z-2)}$ in $\{1<|z|<2\}$ and $\{2<|z|<\infty\}$

I'm trying to get the Laurent Series expansion of the function stated in the title in the stated regions. My approach is as follows: We can first break up $f(z)$ using partial fractions ...
0
votes
1answer
24 views

If $x_k ≥ 0\;\forall \in \mathbb N$, and $y_k$ a bounded sequence, then the series $\sum_{k=1}^\infty x_ky_k$ converges

Hi I'm really struggling with this proof. For a start I'm struggling to believe it's true: For example, if we take $x_k = \dfrac{1}{k^2}$ and $y_k = -k^3$ (which is bounded above by any positive ...
4
votes
3answers
101 views
+100

the first $2k$ terms of the power series of $\sec x + \tan x$ at $x=-\pi/2$

We know the power series of $\sec x+\tan x$ is as follows, $f(x)=\sum_{n\geq 0}\frac{E_n}{n!}x^n$, where $E_n$ is Euler Zigzag numbers and clearly the radius of convergence of $f(x)$ is $\pi/2$. ...
0
votes
1answer
40 views

Sum of Series - Intelligent Manipulation

I have been learning about sums of series, and am very curious: If we know that $e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$ What is the value of the following power series: ...
0
votes
2answers
22 views

Formal power series question

$$(1-t)^d \sum_{k = 0}^{\infty} \binom{d+k-1}{d-1} t^k = 1$$ How can this be proven? Thanks in advance.
0
votes
1answer
39 views

Maclaurin series construction

I am asked to find the Taylor (Maclaurin) series for $9xe^x$ at $x=0$. I did the following: $f(x)=9xe^x \implies f'(x)=9e^x(1+x) \implies f''(x)=9e^x(2+x)$ et cetera. This yields: $P_0(x)=0, ...
0
votes
0answers
10 views

How to determine singularities of a series?

Given a double Fourier series, how do we determine its singularities ? PS: I wonder how we find singularities(mathematically) if a function cannot be expressed in a closed form.
2
votes
0answers
51 views

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent. I use a root test: $\lim_{n\rightarrow\infty} |\frac{e^{nz^2}}{n}|^{1/n}=\lim_{n\rightarrow\infty} ...
-1
votes
1answer
45 views

Coefficeient of $x^k$ in $(1+x)^n$ when $n<0$

I know this is a very basic question. But I simply cannot derive the final answer. We have the alternate form of binomial theorem if we want to deal with negative exponents. ...
0
votes
0answers
15 views

To find sum of series involving combinations and show convergence of series

to make sum of series including combinations ${N\choose 1}{N\choose 0}+{N \choose 2}{N\choose 1}a^2 b^{-2} + {N\choose 3}{N \choose 2}a^4 b^{-4}+{N\choose 4}{N \choose 3}a^6 b^{-6}+...$Is it possible ...
0
votes
1answer
49 views

Solving the ODE $y''-2x^2y'+4xy=x^2+2x+2$ using power series

I am trying to solve this nonhomogeneous ODE: $$y''-2x^2y'+4xy=x^2+2x+2$$ I know it's a power series, but when I get down to the very end, I end up with a $C_0$ term, a $C_1$ term, and a $C_2$ term. ...
0
votes
0answers
35 views

What are the possible expansions for $f(z)$ at $0$ for disks and annuli?

For the expression $f(z)$ what are its all possible expansions (I am considering disks and annuli) around the origin and where do they converge? $$ f(z) = z + 2z^2 + 3z^3 + \ldots + nz^n + \ldots = ...
0
votes
0answers
15 views

Large-z limit of the *other* second derivative of the Laguerre polynomial

I'm trying to find the asymptotic behavior of the second derivative of the Laguerre polynomial (more precisely, the associated analytic function), $\frac{\partial}{\partial \nu^2}L_{\nu}(z)$, as $z\to ...
1
vote
2answers
43 views

Estimating integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$

Estimate the definite integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$ with an error of at most $10^{-4}$, using the Alternating Series Estimation Theorem. My approach is as follows: I found the ...
0
votes
2answers
43 views

Power Series: Derivative

Given a Banach space $E$. Consider a series: $$|t|\leq R:\quad\sum_{k=0}^\infty A_k t^k\quad(A_k\in E)$$ Is there an elegant proof of: $$\left(\sum_{k=0}^\infty A_k t^k\right)'=\sum_{k=0}^\infty A_k ...
0
votes
0answers
17 views

Find power series expansion and radius of convergence

I have to find power series of $\frac{1}{(z-1)(z-2)}$ centered at $3+i$ and give its radius of convergence. I just simply transformed it using partial fractions and geometric series expansion and ...
1
vote
4answers
94 views

How do I express the sum $(1+k)+(1+k)^2+(1+k)^3…+(1+k)^N$ for $|k|<<1$ as a series?

Wolfram Alpha provides the following exact solution $$ \sum_1^N (1+k)^i = \frac{(1+k)\,((1+k)^N-1)}{k}.$$ I wish to solve for $N$ of the order of several thousand and $|k|$ very small (c. $10^{-12}).$ ...
0
votes
1answer
21 views

Find Laurent series for $\frac{(z+1)}{z(z-4)^3}$ in $0<|z-4|<4$

Find Laurent series for $\frac{(z+1)}{z(z-4)^3}$ in $0<|z-4|<4$. First we perform partial fraction and we get: $\frac{A}{z}+\frac{B}{z-4}+\frac{C}{(z-4)^2}+\frac{D}{(z-4)^3}$. My first ...
1
vote
1answer
20 views

Find infinite set for which the series diverges

I'm looking to clarify the meaning of a question, and would greatly appreciate any feedback. Given a function $f_n(x)$, I am to construct an infinite set S such the series ...
0
votes
0answers
17 views

Generating Functions and Power Series

The question is Use partial fractions and the generalised binomial theorem to write $R(x)$ as a power series where $R(x)$=$-1+5x\over{1-x-2x^2}$ I found the partial fractions to be ...
3
votes
3answers
40 views

Power series expansion involving non integer exponent

I'm working on a real and complex analysis course right now and one power series question has me really stumped: I'm not sure what to do with the non integer in the exponent, as my initial plan of ...
0
votes
1answer
27 views

Writing ODE solution as sun of 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 ...
1
vote
1answer
25 views

For which values of x does the power series converge or diverge?

First, I know that the series converges when |x+2| < R and diverges when |x+2| > R. So now I have that the radius of convergence is somewhere between 2 and 3. However, this doesn't really give me ...
1
vote
3answers
121 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
votes
1answer
16 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
vote
0answers
19 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
votes
1answer
21 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
votes
3answers
41 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
votes
2answers
23 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
votes
1answer
26 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 ...
1
vote
1answer
29 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
votes
1answer
24 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
votes
1answer
33 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
votes
4answers
289 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
vote
1answer
42 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
votes
1answer
48 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
votes
3answers
95 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
votes
2answers
68 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
votes
2answers
40 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
votes
1answer
171 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
44 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
votes
1answer
32 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
vote
1answer
22 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
votes
3answers
33 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
34 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 ...