1
vote
0answers
15 views

Rational Series VS Algebraic Series

I am reading a paper on combinatorics. It mentions some generating functions are rational series and others are algebraic series. I do not understand the difference, can someone help? EDIT $1$: The ...
2
votes
2answers
78 views

Convergence of the series $\sum\limits_{n\geqslant1}(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n})$

If $x >0$, consider the series $\sum\limits_{n=1}^\infty a_n$, where $$a_{n}=(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n}).$$ Is it convergent? This question is in my textbook. I don't know how to ...
-1
votes
0answers
45 views

Infinite radius of convergence [on hold]

If the root test limit tends to infinity, is that sufficient to say we have an infinite radius of convergence? Thanks
2
votes
1answer
73 views

Power series difficulty

How would I find the region of convergence of the series of $\frac{1}{n^3}(\frac{z+1}{z-1})^n$. I thought about rewriting $\frac{z+1}{z-1}$ as $\frac{2}{z-1}+1$ but I don't think that helps. Thanks
0
votes
2answers
25 views

Radius of convergence query

Find the radius of convergence of the series of $\frac{2^n(4z-8)^n}{n}$ My answer: $(4z-8)^n=4^n(z-2)^n=2^{2n}(z-2)^n$. Let $c_{n}=\frac{2^{3n}}{n}$. Then $\frac{c_{n}}{c_{n+1}}=\frac{n+1}{2n}$ so ...
0
votes
2answers
27 views

Find power series representation of $ x/(x^{2}+9)^{2}$

I'm not sure how to do it since the entire bottom term is squared. Is there a geometric series I should use? Or differentiation?
2
votes
2answers
38 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then find whether the above series ...
1
vote
2answers
68 views

If a series converges then the power series converges for all z

How can I prove that if $\sum \limits_{n=1}^{\infty} c_n$ , $c_n\in \mathbb{C}$, converges then $\sum \limits_{n=1}^{\infty} c_n \frac{z^n}{1-z^n}$ converges for all z in $\mathbb{C}$ with ...
0
votes
0answers
52 views

sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$

Let $x_n$ be a sequence of real numbers such that $x_n\in(0,1).$ Find the sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$ My answer is $\frac{(x_n)'}{(1-x_n)^2}.$ But the term $(x_n)'$ make me ...
0
votes
0answers
12 views

How to prove a function $f(t)=\sum_{n=0}^{\infty} a_n t^{2n}$ with real coefficients $a_n$ is regular in the strip $-\pi/4\le \Im(t) \le \pi/4$?

How to prove a function $f(t)=\sum_{n=0}^{\infty} a_n t^{2n}$ with real coefficients $a_n$ is regular in the strip $-\pi/2\le \Im(t) \le \pi/2$? What will be the sufficient conditions on the real ...
1
vote
0answers
12 views

Taylor's expansion of the singular part of an analytic function

Assume $f$ is analytic on the annulus $R_1<|z-a|<R_2$. Assume $R_1<r<|z-a|$. Define $f_2$ by $$f_2(z)=\frac1{2\pi i}\int_{|x-a|=r}\frac{f(x)dx}{x-z}$$ $f_2$ is analytic on $|z-a|>r$. ...
4
votes
3answers
53 views

Radius of $\sum a_n b_n x^n$ via radii of $\sum a_n x^n$ and $\sum b_n x^n$

Series $\sum a_n x^n$ and $\sum b_n x^n$ have radii of convergence of 1 and 2, respectively. Then radius of convergence R of $\sum a_n b_n x^n$ is 2 1 $\geq 1$ $ \leq 2$ My ...
3
votes
1answer
44 views

$\sum_{n=0}^{\infty} a_n x^n$ and $\sum_{n=0}^{\infty} a_{n^2} x^n$ with different radii of convergence

Could you give an example of $$\sum_{n=0}^{\infty} a_n x^n$$ and $$\sum_{n=0}^{\infty} a_{n^2} x^n$$ that have different radii of convergence?
1
vote
0answers
62 views

Can there be a power series with interval of convergence $[k, \infty)$?

My answer : NO Because Interval of convergence is of the form $(a-R, a+R)$ Where $a$ is centre of convergence. If there exists a power series with Interval of convergence $[k, \infty)$ $ $ We ...
2
votes
2answers
97 views

Complex series radius convergence

How to find the values for which $z$ converges, $z\in\mathbb{C}$, in the serie $$\sum_{n=1}^{\infty}\frac{1}{(1+|z|^{2})^{n}}$$ I know I have to use the convergence radius expression, but what I ...
1
vote
2answers
179 views

Solution to curious infinite series

How exactly does one find a closed form to: $$ \sum_{i=0}^{\infty}\left[\frac{1}{i!}\left(\frac{e^2 -1}{2} \right)^i \prod_{j=0}^{i}(x-2j) \right]$$ When expanded it takes on the form $$1 + ...
-1
votes
3answers
74 views

Can there be more than one power series expansion for a function.

I guess the answer is NO, for polynomials. I know that there are more than one series expansion for every function. But I am talking about power series here. All Ideas are appreciated
0
votes
0answers
22 views

Sum of analytic functions

The sum of analytic functions is analytic. Does it mean that: $\sum_{i}\sum_{n=0}^{\infty}a_{in}x_{i}^{n} = \sum_{n=0}^{\infty}a_{n}x^{n}$ ? Is this also true $\sum_{i}\sum_{n=0}^{m}a_{in}x_{i}^{n} ...
8
votes
2answers
129 views

Convergence of differentiated power series

Consider $\displaystyle f(z)=\sum_{k=0}^\infty a_k z^k$ and suppose that $\displaystyle\sum_{n=0}^\infty f^{(n)}(0)$ converges. Prove that $\forall z \in \mathbb C, ...
4
votes
0answers
80 views

Is there a known evaluation of $\sum_{k=1}^{\infty} \frac{1}{k^k}$? [duplicate]

Is there a known evaluation of $$\sum_{k=1}^{\infty} \frac{1}{k^k}$$ Wolfram Alpha says that it converges to $\approx 1.29129$.
2
votes
0answers
48 views

How to power series expand determinants?

Say $g$ is a ($d\times d$) matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same ...
0
votes
1answer
18 views

Find the radius of convergence of $\sum_{n=0}^{\infty} (3^n + (-5)^n) x^{7n}$

I have to find the radius of convergence of the series $$\sum_{n=0}^{\infty} (3^n + (-5)^n) x^{7n}$$ I know that I will have something like $|x^7|<\frac{1}{L}$. I tried finding $R$ with ...
0
votes
1answer
37 views

Taylor Series expansion and first four terms of $7x^2 e^{-4x}$

As the series I got $$ \sum_{n=0}^\infty (-1)^n(4x)^n/n! $$ which I think is right. However, I am not sure how to get the first four non zero terms.
1
vote
2answers
30 views

Meaning of interval of convergence when approximating functions

Let's say I have a Taylor series approximation, $p(x)$, of a function $f(x)$ at $a$: $$ p(x)=\sum_{n=0}^\infty{\frac{f^{(n)}(a)}{n!}(x-a)^n} $$ And that this Taylor series has a radius of ...
1
vote
0answers
45 views

Sum of a power series

I have to find the sum of this series $$\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n!}$$ Using integral, I got $$\int\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n!}=\sum_{n=1}^{+\infty}\frac{x^{n}}{n\cdot n!}$$ I ...
1
vote
1answer
19 views

Convergence of series, using big oh or little oh notation.

Let $p\in \mathbb{R}$ and $a_n=(e-(1+1/n)^n)^p$. For which $p$ will $\sum_{n=1}^{\infty} a_n$ converges? Because of the "additive look" of $a_n$, I tried to use taylor expansion and big oh, little oh ...
0
votes
3answers
77 views

Represent $\frac1{(1+x)^2}$ as a power series

Represent $\frac1{(1+x)^2}$ as a power series. if we put $\frac1{1+2x+x^2}$: $$ \sum _{n=1}^{+\infty}\left(-x-x^2\right)^n $$ and if we differentiate $\frac{-1}{1+x}$: $$ -\sum ...
1
vote
3answers
57 views

calculate radius of convergence

Let $\{a_n\}_{n=0}^{\infty}$ be sequence such that $$a_1 = a_0 = 1$$ $$a_{n+1}=a_n+ a_{n-1}$$ show that the radius of convergence of $\sum\limits_{n=0}^{\infty \:}a_nx^n$ is ...
1
vote
0answers
60 views

How to generilize the the following summation.

While searching for a summation formula I come accross the following equation on wikipedia Equation $$\sum\limits_{k=1}^{n}{k^m z^k}=\left(z\frac{d}{dz}\right)^m\frac{z-z^{n+1}}{1-z}$$ So I tried to ...
10
votes
6answers
3k views

Why infinity multiplied by zero was considered zero here?!

I watched an online video lecture by some professor and she was solving a convergence problem of the power series $$\sum_{n=1}^\infty n!x^n,$$ i.e., she was finding the values of $x$ for which this ...
3
votes
2answers
121 views

Calculate the value of $\sum\limits _{n=1}^{\infty }\:\dfrac{n}{2^n}$ [closed]

In a previous question it is asked to represent $f(x)=\dfrac{x}{1-x^2}$ as a power series. It gave me $\displaystyle\sum _{n=1}^{\infty \:}x\left(2x^2-x^4\right)^{n-1}$. Then they ask to use the last ...
1
vote
0answers
28 views

Rational approximation or series expansion of $K_0$ and $K_1$ for small z

I'm looking for a series expansion of the modified Bessel functions of second kind $K_0(z)$ and $K_1(z)$ for $$|z|<5, ~~|phase(z)| < \pi$$ My $z$ can be described as $z = a\cdot \sqrt{ix}$, ...
4
votes
2answers
45 views

Suppose that $f(x)=\sum_{n=0}^{\infty }c_n{x^{n}}$ for all x

Suppose that $f(x)=\sum_{n=0}^{\infty }c_n{x^{n}}$ for all x Show that if $f$ is an odd function, $c_{0}=c_{2}=...=0$ and if $f$ is an even function, $c_{1}=c_{3}=...=0$
3
votes
4answers
94 views

Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $

Evaluate: $$\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $$ I rewrote the sum as $$\sum_{n=0}^{\infty} \frac {1}{4^{8n-7}(8n-7)} - \sum_{n=0}^{\infty} \frac {1}{4^{8n-3}(8n-3)}$$ Now, I ...
1
vote
2answers
80 views

radius of convergence $\displaystyle \sum_{n=1}^{\infty}n! x^{n!}$

I just wondering radius of convergence following series $$ \sum_{n=1}^{\infty}n! x^{n!} \\ $$ My 1st attempt is 'root test' $$ \sqrt[n!]{|a_{n!} |} =\sqrt[n!]{|n! |} =\sqrt[t]{t} \rightarrow 1 $$ So, ...
-1
votes
2answers
23 views

Find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$

Let $\alpha$ and $\beta$ be two given constants, how to find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$, where $c_0 = {\alpha}^{\beta}$.
0
votes
3answers
27 views

Calculate the area of region between curve f and x axis using series

Consider $f(x)=\sqrt{1+x^4}$ I need to approximately calculate the area of a region between a curve $f$ and the x-axis on [0,1]. However, I need to do this using the five first term non-null of the ...
3
votes
0answers
69 views

Series that diverges at infinitely many points on the unit circle

Initial problem Given $A=\{\alpha_1,\dots,\alpha_k\}$ with $|\alpha_i|=1$, does there exist a power series $\sum a_nz^n$ that converges everywhere on the unit circle except when $z\in A?$ ...
1
vote
2answers
31 views

a series derived from a holomorphic function converges implies that the coefficients converge to $0$

Let $D=\{z\in\mathbb{C}\mid |z|<2\}$. Let $f:D\setminus\{\frac{i}{2}\}\longrightarrow \mathbb{C}$ be holomorphic with $f(z)=\sum_{n=0}^\infty a_nz^n$ for any $|z|<\frac{1}{2}$. Suppose $a_n\neq ...
0
votes
2answers
69 views

Radius of Convergence…very tricky question using very little information

Even one of the maths gurus at my university struggled to get a proof out for this... so I'm almost completely lost! This is the question: Let $a_{n}$ be a sequence of positive real numbers for which ...
0
votes
0answers
64 views

Taylor theorem remainder term

I'm having trouble applying the formula for the remainder in the Taylor's theorem. From Wikipedia we know that for $f(x)=f(a)+f'(a)(x-a)+…\frac{f^{(n)}(a)}{n!}(x-a)^{n}+R$ the remainder $R$ in the ...
0
votes
1answer
14 views

Some questions about series and generalized integrals

Do someone of you know any good site about generalized integrals and series (convergence and divergence) with example exercises with detailed solutions ? I would be happy if you did know any. My ...
2
votes
1answer
41 views

Two questions about this solution /proof

Consider the following theorem: If $\sum_{n=0}^\infty a_n x^n $ converges for all $x \in (-R,R)$ then the differentiated series $\sum_{n=0}^\infty n a_n x^{n-1}$ converges for all $x \in (-R,R)$. ...
3
votes
0answers
57 views

Proof of Abel's theorem

I tried to prove: If $g(x) = \sum_{n=0}^\infty a_n x^n$ is a power series that converges at $x= R > 0$ then it converges uniformly on $[0,R]$. Please can you check my proof? Let $\varepsilon ...
1
vote
0answers
24 views

How do we show $\ln z=\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}(z-1)^n$ for all $z\in\mathbb{C}$ with $|z-1|<1$?

Let $$g:B_1(1):=\left\{z\in\mathbb{C} :|z-1|<1\right\}\to\mathbb{C}\;,\;\;\;z\mapsto\ln z-\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}(z-1)^n$$ (1) In a first step, I'm asked to show, that $g$ is ...
1
vote
2answers
53 views

writing $\ln(1+x)$ as power series

\begin{align*} \left[\ln\left(1+x\right)\right]' &= \frac{1}{1+x}\\ &= \frac{1}{1-(-x)}\\ ...
0
votes
3answers
42 views

Skip terms in power series of $\cosh$ and $\sinh$

Is there a way to skip every second term in the power series representation of $\sinh{x}$ and $\cosh{x}$ and adjust the other terms accordingly (approx.)? So, instead of $$\sinh{x} \approx x + ...
1
vote
2answers
49 views

Use power series $\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ to show $\sum^{\infty}_{n=1} \frac {1} {n(n+1)} =1$.

Consider $\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ (power series). I've found that the sum-function $g(z) := \sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ is defined and continuous on $|z| \le ...
0
votes
1answer
18 views

Let $R$ be radius of convergence for $\sum^{\infty}_{n=0} a_n z^n$. find radius of convergence for $\sum^{\infty}_{n=0} a_n z^{kn+l}$ in terms of $R$.

Let $R \in [0,\infty)$ be radius of convergence for $$\sum^{\infty}_{n=0} a_n z^n$$. For $k \in \mathbb N, l \in \mathbb N_0$ find radius of convergence for $$\sum^{\infty}_{n=0} a_n z^{kn+l}$$ in ...
1
vote
1answer
22 views

Let $\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ be a power series. Show sum-function $g(z)$ is continuous on $|z|\le 1$.

Let $\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ be a power series. I've shown that radius of convergence is $R=1$. I've a theorem saying that the sum-function $g(z)=\sum^{\infty}_{n=1} \frac ...