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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0
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0answers
51 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
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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?
6
votes
2answers
189 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
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0answers
23 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
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0answers
136 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
46 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
45 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
66 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
72 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
46 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
25 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
59 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
43 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
76 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
35 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
13 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
55 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
30 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
83 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
37 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
38 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
117 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
29 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
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0answers
39 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
49 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
57 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
190 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
94 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
78 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
162 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
125 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
56 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
49 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
58 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 ...
1
vote
0answers
20 views

Logarithm of the basic Lubin-Tate formal group

Let $K$ be a local field with finite residue field of cardinality $q$. Let $\pi$ be a uniformizer. The basic Lubin-Tate group (associated to $\pi$) is the unique formal group associated to the ...
8
votes
2answers
146 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, ...
0
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0answers
29 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
4
votes
0answers
85 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 $\approx 1.29129$.
0
votes
1answer
48 views

How to derive the formula to calculate the amount of cubes in a pyramid?

The pyramid looks like: For which I managed to derive the formula for the count of cube sides (ignoring the top). This was easy by simply thinking about it as a triangle: If we have 4 squares wide ...
2
votes
2answers
56 views

Convergence radius of power series is infinite

Which function is given by a power series whose convergence radius is infinite? $$A. \ \ \ e^{-\frac{1}{x^2}}$$ $$B. \ \ \ \sin{\left(\frac{1}{x}\right)}$$ $$C. \ \ \ ...
0
votes
0answers
32 views

Convergence of two power series

I just wanted to know, whether my results are correct. I should find the radius of convergency in both cases: $\sum_{n=1}^\infty \frac{z^{2n}}{n^23^{n}}$ with a quotient criterion ...
2
votes
0answers
52 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 ...
1
vote
2answers
111 views

Using series find $\int_0^1 \sqrt{1+x^4}\hspace{1mm} dx$ up to $2$ decimal places

I cannot figure out an aesthetic way to do this. Can someone give a beautiful solution to this ugly question? This is what I have tried yet. I used the fact that $$x = ...
0
votes
1answer
37 views

Radius of convergence of entire function

Let $f$ be an entire function on the complex plane. Is the radius of convergence of $f$ around any point $z_0$ infinite? If so, why? Thank you.
0
votes
1answer
19 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 ...