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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2
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3answers
54 views

meromorphic function with a pole on the unit circle diverges

Let $f$ be a meromorphic function in a neighborhood of the closed unit disk $\bar{\mathbb{D}}$. Suppose that $f$ is holomorphic in $\mathbb{D}$ and $$ f(z) = \sum_{n=0}^\infty a_n z^n $$ for $z \in ...
1
vote
1answer
42 views

Finding radius of convergence using root test

Find the radius of convergence of the following power series $$\sum_{n=1}^{\infty} \frac{2^n + 1}{n} x^n.$$ Using the ratio test, I have found that the radius of convergence is $R = \frac{1}{2}$. I ...
1
vote
1answer
44 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 ...
1
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1answer
42 views

Number of ways distribute 12 identical action figures to 5 children

Need a little help with this problem. Use generating functions to determine the number of different ways 12 identical action figures can be given to five children so that each child receives at most ...
2
votes
2answers
84 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 ...
2
votes
2answers
60 views

Different Definitions Of The Sine Function

I was hoping someone could give me a flow chart or high-level map connecting all of the definitions of the sine function, with some of the reasons why we care next to each. I've tried this but I'm not ...
0
votes
2answers
37 views

Solution of the Legendre's ODE using Frobenius Method

This is the Legendre's differential equation given in my book: $(1-x)^{2}\ddot{y}-2x\dot{y}+k(k+1)y=0$ I solved this equation by taking: $y=x^{c}\{a_{0}+a_{1}x+a_{2}x^{2}+.....+a_{r}x^{r}+.....\}$ ...
2
votes
1answer
76 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
28 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 ...
1
vote
0answers
26 views

What is the power series for a half-exponential function?

What is the power series of a half-exponential function? Half-exponential means that $f(f(x)) = y^x, y > 1$
1
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3answers
85 views

Power Series Convergence comparison

Given $\sum_{n=0}^\infty c_n4^n$ is convergent, can this be used to find the convergence of $\sum_{n=0}^\infty c_n(-2)^n$?
0
votes
0answers
30 views

Confused by a Laplace transform of $f(t)=t^ne^{at}$

Having looked at my lecture notes I was confused by the following part of a derivation of a Laplace transform for the function $\;f(t)=t^ne^{at} ,\quad n\ge0,\; a \in \mathbb{C}, \; f(t)=0 \;\forall ...
0
votes
2answers
29 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?
5
votes
1answer
151 views

Prove $(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j} $

I stumbled upon the identity $$(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j}. $$ The right-hand side is a polynomial. ...
0
votes
0answers
19 views

Vanishes of second order

I encountered the following statement in an article: Let $f(X) = f(X_1, ..., X_N) = \sum_{|\alpha| \geq 2} {a_{\alpha}X^{\alpha}}$ be a convergent power series which vanishes of second order at ...
5
votes
0answers
131 views

Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
6
votes
1answer
41 views

Divergence set at radius of convergence

I came up with this question on my own while I was musing around reviewing notes. After unsuccessful Google search (thwarted by a deluge amount of webpages on basic calculus), I decided to ask here. ...
2
votes
2answers
41 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
65 views

Find a power series for function

I'm having some difficulty with this problem even while noting the hint. I expressed the function as $(1/2)\frac{1}{1-(-3x/2)}$ and then thought I would work with $1/2$ of the infinite sum of ...
1
vote
2answers
71 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
55 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 ...
2
votes
0answers
42 views

Origin of Laplace Transform

Is the Laplace transform the continuous version of the infinite power series? $$ \sum_{n=0}^\infty a_nx^n$$ becomes $$\int_0^\infty f(t)e^{-st}dt$$ I learned this by watching this video lecture: ...
2
votes
1answer
21 views

Application of Rouché: Equality of a power series and a finite series

Let $f(z) = \sum_0^\infty{a_n z_0^n}$ be a complex power series with radius of convergence $R>0$ and let $z_0 \epsilon \, \mathcal{U}_R(0)$ an arbitrary point. I need to show with $Rouché$ : For ...
0
votes
1answer
57 views

Hypergeometric function representation

Is it possible to express the following sum in terms of the hypergeometric function $_2F_1$: $$ f(x) = \sum_{n=0}^\infty\frac{(-ax)^n}{n!~\Gamma(b-n)} $$ with $a$ and $b$ constant values ($x>0$ ...
2
votes
2answers
41 views

How to compute power series by composition

Is it possible to compute the power series of every function (e.g. around $0$) just by composing of the power series of its arguments? For example: The power series of $\sin(x^2)$ around $0$ is the ...
0
votes
1answer
74 views

how I could show that: $\sum_{k=0}^n\binom{n}{k}(-1)^k\frac{n^2+n+1}{nk+n+1}\le1 $?

Show that for all positive integers $ n $ : $\sum_{k=0}^n\binom{n}{k}(-1)^k\frac{n^2+n+1}{nk+n+1}\le1 $ I would be interest for any replies or any comments
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
26 views

Abel's Theorem, alternate proof

I'm trying to solve: Suppose $\sum_{n=1}^\infty a_n$ converges. Prove that: $$ \lim_{r\to1^-}\sum_{n=1}^\infty r^n a_n = \sum_{n=1}^\infty a_n. $$ Hint: Sum by parts. In class, I have seen a ...
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 ...
0
votes
1answer
22 views

Laplace transform and Power series

As it can be read here, Discrete to Continuous Representations of Functions via Laplace Transforms? the Laplace transform is a continuous analog of a power series in which the discrete parameter n is ...
-3
votes
1answer
80 views

Compute the sum of the power series in a closed form

Please help me to compute the sum: $$\sum_{n=1}^{\infty } \frac{n!}{n^{n}} x^{n}$$ in a closed form. === here ends the original post. After a few minutes I've added the following information: ...
1
vote
1answer
26 views

A holomorphic function $f$ has an essential singularity in $0$ iff $\exists(z_k)_k$ s.t. $z_k\to 0$ and $|z_k^mf(z_k)|\to\infty$ for all $m$

Let $f:\mathbb{C}\setminus\left\{0\right\}\to\mathbb{C}$ be a holomorphic function $\Rightarrow$ $f$ has an essential singularity in $0$ if and only if $\forall m\in\mathbb{N}:\exists ...
3
votes
1answer
45 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
votes
1answer
34 views

How do you represent f(x+h) and f(x−h) as a Taylor series using the taylor series formula?

I know the answers are below, however i am not quite sure what to substitute as the "a" in the Taylor series formula. $f(x+h)=f(x)+f′(x)⋅h+\frac 12f′′(x)\cdot h^2+\cdots+\frac 1{n!}f^{(n)}(x) \cdot ...
1
vote
0answers
63 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
104 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 ...
0
votes
1answer
27 views

$\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor\log_2n\rfloor}}{\lfloor\log_2n\rfloor+1}(x-x_0)^n$ convergence/divergence

I have a problem with determining whether these series are convergent/divergent at the endpoints of their radii of convergence. None of the tests or approaches I know seems to by applicable here... ...
1
vote
0answers
25 views

Power series solution for a DE with Frobenius method

The given DE is $(x²-3)y"+2xy'=0$ Since there is a singular point ($x=\pm\sqrt{3}$) I used the Frobenius method. I found two indicial relationships: $-3r(r+1)=0$ and $-3(r+1)(r+2)=0$ because I have ...
1
vote
1answer
43 views

Finding the power series representation for $\ln(1 -10x)$ via integration.

I'm trying to find the power series representation for $ \ln(1-10x) $ Attempt at solution: $$ \ln(1-10x) = \int {-10\over1-10x} \ dx = -10 \int \sum_{n=0}^\infty (10x)^n dx $$ $$ = -10 ...
2
votes
1answer
34 views

Convergence of a Complex Power Series at the radius of convergence

I am currently reviewing some complex analysis, and have come across this question which I absolutely have no idea on how to attempt: Suppose the radius of convergence of the power series $f(z) = ...
2
votes
2answers
186 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 + ...
0
votes
1answer
89 views

If $f(z):=\sum_{n=0}^\infty a_nz^{-n}$ is compact convergent, then $f$ is holomorphic

Let $\left(a_n\right)_{n\in\mathbb{N}}\subset\mathbb{C}$ such that $$f(z):=\sum_{n=0}^\infty a_nz^{-n}$$ is compact convergent on $B_r(0)\setminus\left\{0\right\}$. I want to show: $f$ is ...
0
votes
3answers
26 views

Let $S(x)=\sum\limits_{k=0}^\infty \frac{(-1)^k x^{2k+1}}{(2k+1)!}$. Define $C(x):=S'(x)$ and show that $C'(x)=-S(x)$.

so this problem I am trying to solve says Let $S(x)=\sum\limits_{k=0}^\infty \frac{(-1)^k x^{2k+1}}{(2k+1)!}$. Define $C(x):=S'(x)$ and show that $C'(x)=-S(x)$. Well, I get that ...
-1
votes
3answers
77 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
6
votes
2answers
152 views

How to show that $\lim\limits_{n\to\infty}n^{2/3}a_{n}=\sqrt[3]{2}/\Gamma{(1/3})$

Let $$\left(\dfrac{1+x}{1-x}\right)^{1/3}=\sum_{n=0}^{\infty}a_{n}x^n,|x|<1$$ Show that $$\lim_{n\to\infty}n^{2/3}a_{n}=\dfrac{\sqrt[3]{2}}{\Gamma{\left(\dfrac{1}{3}\right)}}$$
7
votes
1answer
120 views

Find the power series of $x\ln(1-x)$.

So the exercise I had to do was: Find the power representation of $x\ln(1-x)$. The way to go was finding the power series representation of $\ln(1-x)$ and then multiply it with $x$. But why can't you ...
0
votes
0answers
49 views

closed form expression for an infinite series

Is there any closed form expression for the infinite sum $∑_{n≥0}q^{n(n+1)/2}(1+q)(1+q^2)...(1+q^n)u^n$ where both q and n are variables and $n \in N∪0$?
0
votes
1answer
46 views

find radius of convergence

Suppose the radius of convergence of $\sum_n a_n x^n$ is $r$ ($r$ is a positive number). Prove that the radius of convergence of $\sum_n a_n^2 x^n$ is $r^2.$ I've tried to use Cauchy–Hadamard ...
3
votes
2answers
48 views

Show if the series $f(x)=\sum\limits_{k=1}^\infty \frac{1}{k} \sin(\frac{x}{k})$ converges uniformly or not.

So this is part of a different problem. The book and my professor say that the series $f(x)=\sum\limits_{k=1}^\infty \frac{1}{k} \sin(\frac{x}{k})$ converges uniformly on [0,1] by the Weierstrass ...