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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2answers
31 views

upper bound for the series $S (x) = \sum_{k=1}^{\infty} \frac{(x_n-n)^k}{k!}$ from $|x_n -(n+1)|\leq x$.

I've been trying to find a tight upper bound for the series $$S (x) = \sum_{k=1}^{\infty} \frac{(x_n-n)^k}{k!}$$ in terms of finite value $x\in \mathbb R$, where: 1- $\{x_n\}$ is a sequence of a ...
0
votes
1answer
34 views

Finding Radius of Convergence when Power Series is not in Standard Form $\sum a_n x^n$

I was working on finding the radius of convergence of $$\sum _{n\ge1} \frac{(-1)^n}{\sqrt n 2^n}x^{n^2} \,\,\,\,\,(*)$$ The radius for this example happens to be $R=1$ since $$R=\frac{1}{\limsup ...
2
votes
3answers
39 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
39 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 ...
7
votes
1answer
610 views

Equation about generating functions and subfactorial

Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following explanation) of variable $w$, try to solve out $G_n(w)$ for all $n\ge0$ from the ...
1
vote
1answer
28 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
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
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0answers
20 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 ...
0
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2answers
23 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
2answers
48 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 ...
-1
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0answers
46 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
26 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
81 views

Power Series Convergence comparison

Given $\sum_{n=0}^\infty c_n4^n$ is convergent, can this be used to find the convergence of $\sum_{n=0}^\infty c_n(-2)^n$?
5
votes
1answer
139 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. ...
4
votes
0answers
98 views
+50

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 ($x$ ...
0
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0answers
27 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?
0
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0answers
18 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 ...
6
votes
1answer
34 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
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
57 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
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
1answer
55 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
1answer
20 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
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 ...
1
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0answers
35 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
2answers
34 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
0answers
45 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 $n \in N∪0$?
1
vote
2answers
180 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
62 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 ...
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 ...
1
vote
0answers
25 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 $|z-a|>r$. ...
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
78 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: ...
2
votes
1answer
3k views

Maclaurin Series for $\arctan(x)$ by successive differentiation

I am trying to find a Maclaurin Series for $\arctan(x)$ up to the term with the fifth power of x and I have to use the method of successive differentiation. I know (from an example in my notes) the ...
1
vote
1answer
20 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 ...
1
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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 ...
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
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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 ...
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 ...
0
votes
1answer
26 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... ...
0
votes
0answers
28 views

Radius of Convergence and the Frobenius Method

Consider the equation $$4xy'' + 2y'+ y = 0$$ I know that $x=4$ is a regular singular point, and in the notation that my uni uses, we say that: $$(x-x_0)^2 y'' + (x-x_0)p(x)y' + q(x)y = 0$$ where ...
1
vote
0answers
24 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
2answers
31 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
29 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) = ...
6
votes
2answers
344 views

What makes a Maclaurin Series special or important compared to the general Taylor Series?

I realize that the Maclaurin Series is a special form of the Taylor Series where the series is centered at $x=0$, but I have to wonder what's special about it such that it deserves its own special ...