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).

learn more… | top users | synonyms

1
vote
1answer
23 views

Translations AND dilations of infinite series

Sometimes, when working with infinite series, it's useful to add "dilated" or "translated" versions of the infinite series, term by term, back to the original. There are ways of making this rigorous ...
1
vote
1answer
78 views

limit of jacobi theta 2 or simple series

I have a simple problem: I need to evaluate the limit $x\rightarrow 1$ of the Jacobi Theta function 2 $$\Theta_2(m,x)=2x^{1/4}\sum_{k=0}^\infty x^{k(k+1)}\cos((2k+1)m)$$ when $m=0$, that to say ...
1
vote
2answers
46 views

Finding the co-efficients of this power series

I am required to find the co-efficients of this power series: $2x\ln(1+2x)$ I approached the problem by considering the $\ln(1+2x)$ part as the integral of $2/(1+2x)$ and applied the geometric series ...
1
vote
1answer
56 views

Derivative of exp with definition of differentiability

Prove with the definition of differentiability that $\exp(z)$ is differentiable in $\mathbb C$ and $(\exp(z))' = \exp(z)$ for all $z \in \mathbb C.$ I tried: \begin{align*} \frac{\exp(z+h) - ...
1
vote
1answer
922 views

Is there a simple proof that a power series can be differentiated term by term?

I know that the derivative of the sum of a power series can be calculated by summing the derivatives of the terms, and that the resulting series has the same radius of convergence as the original. The ...
1
vote
1answer
197 views

Applications of higher powers of trigonometric functions

I am after a reference (book, papers etc) about the practical applications of trigonometric functions raised to higher powers. An example is one that I have been using in my own studies: $\cos^4 ...
1
vote
1answer
556 views

The coefficients in the inverse of unit lower triangular Toeplitz matrix

I have a question about the inverse of lower triangular Toeplitz matrix. There is a matrix $Q$ which can be infinite-dimensional. $$ Q=\left[\begin{array}{ccccc} 1\\ -k_{1} & 1\\ -k_{2} & ...
0
votes
2answers
474 views

Behavior of $\sum_{n=1}^\infty n^{-1}z^n$ on the circle of convergence

Consider the following complex power series :$$\sum_{n=1}^\infty\frac{z^n}{n}$$ The radius of convergence of this series is $1$ and the series is divergent for $z=1$. I want to know what are the ...
0
votes
1answer
164 views

What's the limit of coefficient ratio for a reciprocating power series?

I have a question about the coefficient in the inverse of the power series. Assume $$ f=1-\sum_{i=1}^{\infty}(ck_i)x^i, $$ where $c$ and $k_i$ are positive and $0<ck_i<1$ for any $i>0$. ...
0
votes
2answers
88 views

Trying to revert a series with problematic log term

I'm stuck on a problem which I'm not sure has a solution. I have the first few terms of a series I want to invert, $y(x)=\ln(x)+a_0+\frac{a_1}{x}+\frac{a_2}{x^2}+\cdots$ I know the inverse exists ...
-2
votes
1answer
46 views

Find the interval of convergence for these 3 power series

I believe I need to use the root test and ratio test. I've solved the first two, but I'm not too sure I fully understand how to do these, so was hoping someone else could work them out so I could ...
8
votes
4answers
329 views

Asymptotic behavior of $\sum\limits_{k=1}^n \frac{2^k}{k}$

I'm looking for an asymptotic equivalent of $$\sum_{0 < k \le n} \frac{2^k}{k}$$ as $n \to \infty$. A plausible candidate seems to be $\frac{2^{n+1}}{n+1}$ (WolframAlpha plot, and the intuitive ...
7
votes
1answer
105 views

How to derive this interesting identity for $\log(\sin(x))$ [duplicate]

I saw on SE that: $$\log(\sin x)=-\log(2)-\sum_{n=1}^{\infty}\frac{\cos(2nx)}{n} \phantom{a} (0<x<\pi)$$ This is an extremely useful identity, as it helps solve: $$\int_{0}^{\pi} ...
4
votes
2answers
144 views

the sum of $1-\frac{1}{5}+\frac{1}{9}-\frac{1}{13}+…$

I thought this was the real part of the series: $\sum_{n=0}^\infty \frac{i^n}{1+2n}$, with $i=\sqrt{-1}$. When taking the real part I am left with: $\sum_{n=0}^\infty \frac{\cos(n\pi/2)}{1+2n}$. I ...
4
votes
1answer
140 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 ...
4
votes
3answers
214 views

Power Series $0^{0}$

My textbook explains that the power series: $\sum_{n=0}^{\infty} x^{n}/n!$ converges for $x=0$ because the terms of the series get the value 0. My problem with this argument is the first term, ...
3
votes
1answer
69 views

Is my proof that $\frac{\pi}{4}=\sum\limits_{n\geq 0}(-1)^n \frac{1}{2n+1}$ correct?

Respected All I was trying to prove that $$\sum_{n\geq 0}(-1)^{n} \frac{1}{2n+1}=\frac{\pi}{4}$$ What I tried to show like this. We know $$\frac{1}{1+x^2}=(1+x^2)^{-1}=\sum_{n\geq 0}(-1)^nx^{2n}, ...
3
votes
0answers
51 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
440 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 $$\dfrac{1}{(1-z)^2} = (1+z+z^2+\ldots)(1+z+z^2+\ldots) = 1+2z+3z^2+\ldots.$$ ...
3
votes
4answers
234 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
2answers
634 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
4answers
480 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
7answers
347 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 ...
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 ...
2
votes
2answers
191 views

Solution to ODE using Power Series

I'm currently trying to wrap my head around how to solve an ODE with series. The problem I am working on is this: Find the indicated coefficients of the power series solution about x=0 of the ...
2
votes
1answer
104 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
2answers
36 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: ...
2
votes
1answer
111 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
55 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
1answer
111 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 ...
2
votes
0answers
117 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
74 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
3answers
54 views

A power series from $\frac{x}{9+x^2}$

I need to make power series from $\frac{x}{9+x^2}$, and I don't have any idea how. The only thing I know is how to make power series from $\frac{1}{1-q}$. Thank you!
2
votes
0answers
103 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 ...
2
votes
0answers
130 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
1answer
221 views

Definite Sum of Confluent Hypergeometric involving power function

I find it difficult to evaluate the following definite sum: $$ \sum _{k=1}^K \frac{_1F_1[k,1,x]} {2^k} $$ Thank you for your time
2
votes
3answers
145 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

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
3answers
213 views

Showing that $R(x)$ is a proper subset of $R((x))$ if R is a field

I would like to show that if $R$ is a field, then $R(x)$ is a proper subset of $R((x))$, where $R(x)$ is the ring of rational functions, and $R((x))$ is the ring of formal Laurent series. If $f \in ...
2
votes
2answers
931 views

${a_n}$ series of Fibonacci numbers. $f(x)=\sum_{0}^{\infty}a_nx^n$, show that in the convergence radius: $f(x)= \frac{1}{1-x-x^2}$

I'd really like your help with this following problem: Let ${a_n}$ be a Fibonacci series $a_1=a_0=1$ and $a_{n+2}=a_n+a_{n+1}$ for every $n \geq 0$. Let $f(x)=\sum_{0}^{\infty}a_nx^n$, I need to find ...
1
vote
1answer
135 views

Radius of convergence of a power serise involving the Fibonacci sequence.

Consider the power series $$\sum_{n=0}^{\infty}a_nz^n.$$ where, $a_0=0$ , $a_1=1$ , $a_n=a_{n-1}+a_{n-2}$. Find the radius of convergence of the power series. MY Attempt : Clearly $\{a_n\}$ is a ...
1
vote
0answers
75 views

Infinitely nested radicals

In a recent paper is is stated ( and maybe proved) that we can solve any polynomial equation with nested radicals. Here "nested radicals" means expression such as: $$ ...
1
vote
1answer
31 views

Exponential of a complex number converges absolutely

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$ This converges absolutely for every $z\in \Bbb C$. What does this mean to a layman?
1
vote
1answer
50 views

Finding the coefficients of $h(z)$ laurent series

Consider: $$h(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ Find the coefficients $a_n$ of the Laurent Series of $h(z)$ centered at $z=-2$ I got this from the approach here: Infinite sum complex analysis ...
1
vote
3answers
88 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
1answer
174 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 ...
1
vote
4answers
102 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
78 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
0answers
33 views

Somehow “mirroring” the Taylor-expansion of some $g(x)$

In proceeding with an older discussion of a summation-procedure for divergent series using the matrix of Eulerian numbers I came to a rather general formulation but cannot/do-not-know-how-to make ...
1
vote
1answer
74 views

Moving Center of Power Series

Given a power series: $$\lim_{N\to\infty}\sum_{k=0}^N A_k (z-a)^k$$ I expand the powers: $$\lim_{N\to\infty}\sum_{l=0}^N(\sum_{k=l}^N A_k \binom{k}{l}(-1)^{k-l}a^{k-l})z^l$$ But here I face the ...