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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4
votes
1answer
167 views

Convergence of the power series $\sum \left(\frac{n^n}{n!} x^n \right)$

Find the convergence radius of the serie $$\sum \frac{n^n}{n!}x^n $$ and analyze the absolute convergence and/or uniform. What I've done: It is easy to show that the radius of convergence of this ...
3
votes
1answer
74 views

Show $\sum^\infty (2n-3)!!/(2n)!!$ converges

I would like to compute the sum $\displaystyle\sum_{n=0}^\infty\dfrac{(2n-3)!!}{(2n)!!},$ where the double exclamation point refers to double factorial. Using double factorial identities we get the ...
3
votes
0answers
153 views

Does convergence of power series on radius of convergence imply absolute convergence?

Let $R$ be radius of convergence of power seires $\displaystyle\sum_{k}a_kz^k$. If the power series converges for all $|z|=R$, can we say that it converges absolutely on the radius of convergence? I ...
3
votes
3answers
143 views

Puiseux Series?

WolframAlpha says that $$\sqrt{x^2-1}$$ expanded in Puiseux series near 1 is $\sqrt 2 \sqrt{x-1}$ I don't know what is the Puiseux series, I have search on the net but I don't have understood so much....
3
votes
4answers
548 views

Question Regarding The Power Series For $e^x$

Currently I'm reading Higher Engineering Math by John Bird and under exponential function he talks about obtaining the value of $e$. He begins by saying The value of $e^x$ can be calculated to ...
3
votes
0answers
100 views

On a family of polynomials related to the expansion of $(1+\epsilon)^{x/\epsilon}$ as a series in $\epsilon$

Consider the sequence of polynomials $(P_n)_{n\geqslant0}$ uniquely defined by the recursion $$(P_n)'=\sum_{k=0}^{n-1}\frac{P_k}{n-k+1},$$ valid for every $n\geqslant0$, with the initial conditions $...
3
votes
4answers
730 views

Expansion of $(1-z)^{-m}$

Expand $(1-z)^{-m}$, $m$ a positive integer, in powers of $z$. Since $\dfrac{1}{1-z}=1+z+z^2+\ldots$, we can find $$\dfrac{1}{(1-z)^2} = (1+z+z^2+\ldots)(1+z+z^2+\ldots) = 1+2z+3z^2+\ldots.$$ So, ...
3
votes
2answers
1k views

Formal Proof of Exponential rule

I tried to prove this, but was unsuccessful for a long time.. Any ideas? Prove that $(\exp(x))^y=\exp(xy)$ using the identities, $$\exp(x)=\sum_{n\geq0} \frac{x^n}{n!}, \quad (1+x)^\alpha=\sum_{n\...
3
votes
2answers
921 views

Radius of convergence of product

Let $\sum_{i=0}^\infty a_nz^n$ and $\sum_{i=0}^\infty b_nz^n$ be power series, and define the product $\sum_{i=0}^\infty c_nz^n$ by $c_n=a_0b_n+a_1b_{n-1}+\ldots+a_nb_0$. Find an example where the ...
3
votes
2answers
836 views

Why is Taylor series expansion for $1/(1-x)$ valid only for $x \in (-1, 1)$?

After finding an expansion of $$ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots$$ a quick test of various values for $x$ reveals that this expansion is not valid for $\forall x \in \mathbb{R}-\{1\}$. ...
3
votes
0answers
67 views

Find a series solution to $(x^2-2)y''+6xy'+4y=0$.

Find a series solution to $(x^2-2)y''+6xy'+4y=0$. A. Find the recurrence relation to $a_n$: My answer is $a_{n+2}=a_n\cdot \frac{n+4}{2(n+2)}$ which is correct. B. Using A, write two independent ...
3
votes
4answers
241 views

Show that $\displaystyle\lim_{n\to\infty} nx^n=0$ for $x\in[0,1)$.

Show that $\displaystyle\lim_{n\to\infty} nx^n=0$ for $x\in[0,1)$.
3
votes
7answers
353 views

how to find this generating function

this is the power series: $$\sum_{i=0}^\infty n(n-1)^2 (n-2) z^n.$$ how can I find a generating function from it? I could use the third derivative but the $n-1$ is squared so I don't know what to do....
3
votes
1answer
3k views

Frobenius Method to solve $x(1 - x)y'' - 3xy' - y = 0$

So, Im trying to self-learn method of frobenius, and I would like to ask if someone can explain to me how can we solve the following DE about $ x = 0$ using this method. $$ x(1 - x)y'' - 3xy' - y = 0 ...
2
votes
0answers
187 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 ...
2
votes
2answers
106 views

Branch cut for $\sqrt{1-z^{2}}$ and Taylor's expansion!

I'm working in a problem that involves the equation $$ w(z)=\sqrt{1-z^{2}} \,\, . $$ I already know that there're two branch points in this equation, namely $\pm 1$, so there's a Riemann surface ...
2
votes
3answers
115 views

If $a_0\in R$ is a unit, then $\sum_{k=0}^{\infty}a_k x^k$ is a unit in $R[[x]]$

Let $R$ a ring, and let $$\displaystyle R[[x]]=\left\{\sum_{k=0}^{\infty}a_k x^k\;\middle\vert\; a_k\in R\right\}$$ with addition and multiplication as defined for polynomials. We have that $R[[x]]$ ...
2
votes
1answer
346 views

Complex power series expansion for $f(z) = \frac{e^z}{a-z}$

I have the following homework problem in my complex analysis class: Find the complex power series expansion for the function $$ f(z) = \frac{e^z}{a-z}$$ where $a \in \mathbb{C}$, and $z \ne 0$. I know ...
2
votes
0answers
122 views

Are there other power series for the Lambert W function than this one?

Are there other known power series for the Lambert W function, other than this one: $$W(x) = x-x^2+\frac{3 x^3}{2}-\frac{8 x^4}{3}+\frac{125 x^5}{24}-\frac{54 x^6}{5}+\frac{16807 x^7}{720}-\frac{...
2
votes
3answers
181 views

What is the answer to this limit

what is the limit value of the power series: $$ \lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \frac{x^k}{k^{k-m}}$$ where $m>1$.
2
votes
2answers
78 views

An approximate solution to an ODE

I am interested in the ODE: $x^\prime = x^2 + t^2$ $x(0)=0$ The power-series method is not (easily?) applicable here. Do you have any suggestions how to solve it?
2
votes
0answers
174 views

Irreducible polynomials as formal power series

I'm studing the ring of formal series with complex coefficients $\mathbb{C}[[x]]$. I proved that the polynomial $y^2-x^3-x^2$ is irreducible in $\mathbb{C}[x,y]$ but reducible in $\mathbb{C}[[x,y]]$. ...
2
votes
2answers
79 views

How can i evaluate this power series?

$\sum_{n=0}^{\infty }\frac{1}{2n+1} \left (\frac{1}{3} \right )^{n}\left ( -1 \right )^{n} $ it's solved by power series of arctan. is it possible the answer written by real number?
2
votes
1answer
167 views

Infinitely nested radicals

In a recent paper it was stated (and maybe proved) that we can solve any polynomial equation with nested radicals. Here "nested radicals" means expression such as: $$ \sqrt[n]{a+b\sqrt[n/p]{a+b\sqrt[...
2
votes
2answers
43 views

Analyticity: Uniform Limit

Problem Consider a uniformly bounded sequence over the real line: $$f_n:\mathbb{R}\to\mathbb{C}:\quad|f_n(x)|\leq L$$ Suppose they have analytic continuations with common domain: $$F_n:\Omega\to\...
2
votes
1answer
116 views

What's $\sum{\frac{x^n}{n^3}}$?

What's $\displaystyle f(x)=\sum_{n=1}^\infty{\frac{x^n}{n^3}}$? Note its derivative: $$\displaystyle f'(x)=\sum_{n=1}^\infty{\frac{x^{n-1}}{n^2}}$$ and the next derivative: $$\displaystyle f''(x)=\...
2
votes
1answer
116 views

Evaluate $ \int_0^1 \sum_{k=0}^\infty (-x^4)^k dx = \int_0^1 \frac{dx}{1+x^4} $

I have read this thread and I found in some comments the above named equality. I couldn't follow the transformation, which are done to get from the left to the right side at that point in particular. ...
2
votes
1answer
178 views

If $f$ is analytic, prove that $\overline{f(\overline{z})}$ is also analytic

Let $f$ be an analytic function in an open set $U \subseteq \mathbb{C}$. Let $V=\{z\in\mathbb C:\overline z\in U\}$. Define $g$ on $V$ by $g(z)=\overline{f(\overline{z})}$. Show that $g$ is analytic ...
2
votes
3answers
156 views

Explicit formula for the series $ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $

I was wondering if there is an explicit formulation for the series $$ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $$ It is evident that the converges for any $x \in \mathbb{R}$. Any ideas on a formula?
2
votes
2answers
76 views

Sum of powers of 2 from 1 to log(N).

I came across the following sum: $\sum_{m=1}^{\log_2(N)} 2^{m}$. My intuition tells me that this should be bounded by 2N, but how would I prove this?
2
votes
2answers
162 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
66 views

Does n power of e grow much more faster than its Maclaurin polynomial? [duplicate]

I wonder how to calculate the following limit: $$ \lim_{n\rightarrow\infty}\frac{1+n+\frac{{}n^{2}}{2!}+\cdots +\frac{n^{n}}{n!}}{e^{n}} $$ In the first sight, I think it should be zero, because ...
2
votes
2answers
121 views

What is the expansion in power series of ${x \over \sin x}$

How can I expand in power series the following function: $$ {x \over \sin x} $$ ? I know that: $$ \sin x = x - {x^3 \over 3!} + {x^5 \over 5!} - \ldots, $$ but a direct substitution does not give me a ...
2
votes
3answers
212 views

What is the expression for this summation?

Known that $\sum_{n=0}^{\infty}{x^n}{z^n}=\frac{1}{1-xz}$. If we have $\sum_{n=0}^{\infty}\frac{{x^n}{z^n}}{n\beta + \alpha}$ where $\beta, \alpha $ are element of real numbers but not equal $0$. ...
1
vote
1answer
319 views

using power series expansion to find a holomorphic function which solves a differential equation

Using power series expansions, find a function $f$ which is holomorphic on the unit disk $D:=$ {$z\in\mathbb C:|z|<1$} and solves the differential equation $(1-z^2)f''(z)-4zf'(z)-2f(z)=0$ for $...
1
vote
0answers
39 views

Which functions can be constructed as a limit of monomials of a polynomial?

Most of us know the famous limit $$\lim_{n \rightarrow \infty}\left(1+\frac1n\right)^n = e$$ from elementary calculus. And at some other place (or maybe the same book even) I've learned that $$\...
1
vote
0answers
75 views

series look up site

Is there a site for looking up a series to see some of the associated functions. (In the spirit of Encyclopedia of Integer Sequences OEIS.) In particular I am looking for functions related to $ \sum (...
1
vote
1answer
40 views

Radius of Convergence of Complex Power Series

I need to find the radius of the convergence of $\sum_{n=1}^{\infty}3^{n}z^{n^{2}}$ using the Cauchy-Hadamard formula. I'm not feeling 100% proficient at this method, however, so I'm asking 1) if what ...
1
vote
1answer
238 views

Introduction to Analysis: Multiplication Theorem for Series

I've been stuck on this problem over the weekend so I decided to ask for some direction. The problem reads: "The multiplication theorem for series requires that the two series be absolutely ...
1
vote
2answers
164 views

Identity with Bernoulli numbers: $\sum\limits_{k=1}^{n}k^p=\frac{1}{p+1}\sum\limits_{j=0}^{p}\binom{p+1}{j}B_j n^{p+1-j}$

How I can prove that $$\sum_{k=1}^{n}k^p=\frac{1}{p+1}\sum_{j=0}^{p}\binom{p+1}{j}B_j n^{p+1-j},$$ where $B_j$ is the $j$th Bernoulli number? I hope to find the answer. Thanks for help.
1
vote
1answer
65 views

What is this infinite summation?

We encountered an function defined by the infinite summation as shown below: $$F(x,a):=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{-n}(2a-n)x^{n-1}\Gamma(2a+1)}{a(2a-1)\Gamma(2a+1-n)}$$ Where $$a(2a-1)\...
1
vote
4answers
127 views

Factorial in power series; intuitive/combinatorial interpretation?

It is well known that the terms of the power series of exponential and trigonometric functions often involve the factorial function, essentially as a consequence of iterating the power rule. My ...
1
vote
4answers
168 views

power series for square root matrix

Suppose I have a matrix of the form $$U\ =\ (I+z\thinspace X)^{\frac{1}{2}}$$ where $I$ is the $n\times n$ identity matrix, $z\in\mathbb{C}$ and $X$ is a $n\times n$ arbitrary complex matrix with ...
1
vote
2answers
93 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) $\...
1
vote
3answers
93 views

If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3$, what can be said about convergence at $x=2$ and $x=3$?

Problem: Is the following True or False: If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3 \implies:$ a) $\sum_{n=0}^\infty c_n 2^n$ converges. b) $\sum_{n=0}^\infty c_n 3^n$ converges. ...
1
vote
2answers
332 views

Is $-\log (1-z) = \sum_{n=1}^{\infty}\frac{z^n}{n}$ for $z \in \mathbb{C}, \|z\|=1, z \neq 1$?

Is $-\log (1-z) = \sum_{n=1}^{\infty}\frac{z^n}{n}$ for $z \in \mathbb{C}, \|z\|=1, z \neq 1$ ? In any case, why?
1
vote
1answer
400 views

Calculating the power series expansion about $\pi/2$ of $g(z)=\tan(z/2)$

Calculate the power series expansion about $\pi/2$ of $g(z)=\tan(z/2)$. Now calculate the expansion about $0$. I'm having trouble doing this. I'm not even sure which is the best way to approach it,...
1
vote
3answers
113 views

Formula to $\ln$ that holds on interval $x \geq 1$

In the Wikipedia we can find two formulas using power series to $\ln(x)$, but I would like a formula that holds on the interval $x \geq 1$ (or is possible to calculate $\ln(x)$ to $x \geq 1$ with the ...
1
vote
1answer
31 views

What does 'equating the like-power of $q$' mean?

I am reading a book "Homotopy Analysis Method in Nonlinear Differential Equations" by Shijun Liao chapter 13 Applications in Finance: American Put Options. It is stated there that Substituting ...
1
vote
2answers
78 views

Possible counter example to theorem on Taylor series in complex domain

A well known theorem on Taylor series in complex domain is as follow: Suppose $f(z)$ has Taylor series at $a$ with convergence radius of $R$. Then $f(z)$ has at least one singular point on $|z-a|=R$. ...