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
2answers
1k views

Sum of a power series $n x^n$ [duplicate]

I would like to know: How come that $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$ Why isn't it infinity?
0
votes
1answer
47 views

Power series (representation) of given function

Well I'm wondering if below power series is the correct result of the function - wolfram alpha doesn't give anything like the result. The function $$\frac{x}{2x^2+1} = x\cdot \frac{1}{1- \left ( ...
0
votes
1answer
47 views

Power series convergence question

Does there exist a sequence $c_{n}$ of complex numbers such that $$ \sum_{n=0}^{\infty} c_{n} z^{n} $$ has radius of convergence $R = \infty$, but for all other sequences $c_{n}'$ of complex numbers ...
0
votes
1answer
55 views

Series expansion of quotients

I'd like to start of with a simple formula from a textbook $T(W) = \frac{W^5}{1-2W} = W^5 + 2W^6 + 4 W^7 + \dots + 2^j W^{j+5} + \dots$ Obviously, this is an expansion of the quotient into a power ...
1
vote
1answer
34 views

Expanding One Function in Powers of Another

One sees here that it is possible to expand $f(x) = 2x^3 + 7x^2 + x - 6$ in powers of $x - 2$ by taylor expanding $f(x) = f(x - 2 + 2) = f(2 + h)$ about $2$, and this idea can be used in deriving the ...
5
votes
1answer
68 views

Is $\cos x$ irreducible as a power series?

Let $\mathbb{Q}_{\mathrm{ent}}[[x]]$ be the ring of entire functions with rational coefficients. Is $$ \cos x \;=\; \sum_{n=0}^\infty (-1)^n\!\frac{x^{2n}}{(2n)!} $$ irreducible in ...
2
votes
2answers
93 views

Rate of convergence of binomial series

This is the binomial series: $$(1+x)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k} x^k$$ where $|x|<1$ and $\alpha$ can be a complex number in general. How fast does it converge? I need an upper ...
2
votes
1answer
53 views

Clever way to expand 1/(z^2-n^2) in power series?

Is there a good trick to prove the following identity? $$\frac 1 {z^2-n^2} = -\sum_{i=0}^\infty \frac {z^{2i}} {n^{2(i+1)}}$$ I tried writing out the coefficients as a Taylor series, but this was ...
1
vote
2answers
152 views

$L^p$ spaces and counting measure

currently I am working on the following two exercises as a revision for my exam. Let $\mu$ be the counting measure on $\mathbb N$. Show that if $1 \le p < s < \infty$ then $f \in L^p$ implies ...
0
votes
2answers
93 views

Solving differential equation by using power series.

Find, using the power series: $$y(x)=\sum_{k=0}^\infty a_{k}x^k$$ a solution for the following differential equation: $$y'(x) = -x^2y(x),\,\, y(0)=1$$ What's the convergence radius of the constructed ...
2
votes
1answer
87 views

Why don't power series methods work for linear ODE's with singularities?

My math class tells me power series methods don't work for equations of the form $$f'' +p(x)f' +q(x)f = 0$$ if the functions $p(x)$ or $q(x)$ have singularities at the point about which you're ...
2
votes
2answers
73 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?
0
votes
2answers
46 views

how can I sove approximation evaluation of this integral?

$$\int_{-1}^{0}\sin(e^{x})\,dx $$ approximation of this formula up to difference(error) $1/5000$ Because of the error size $1/5000$ , I think it's solved by taylor expansion.
2
votes
2answers
39 views

Computing the limit of function containing a power series.

Prove that if the sequence $a_{n}$ of real numbers converges to a finite limit; \begin{align} \lim_{n \rightarrow \infty} a_{n} = g, \end{align} then \begin{align} \lim_{x \to \infty} \left({\rm ...
0
votes
1answer
39 views

Ratius convergence of composition of series powers [closed]

For a complex function $f(z) = 1 / (2 + \exp(z))$, I have to find the radius of convergence centered in $1+i$ of the power series $f$
0
votes
2answers
51 views

$ \sum_{n\geq1}(\frac{1}{(n+1)!} ∏_{k=1}^{n} f(k))$ converge or diverge?

Let $f: \mathbb N - {0} \to \mathbb N -{0}$ injective function, verify is the serie $ \sum_{n\geq1}(\frac{1}{(n+1)!} ∏_{k=1}^{n} f(k))$ converge or diverge . I prove for $n=5$ and $n=9$ and found ...
0
votes
1answer
87 views

solve this differential equation using laplace transform and the series method :

Problem : $y''+8ty'-16y = 3 , y(0) = y'(0) = 0 $ I am supposed to use the series method to get F(s) , then get the inverse laplace transform to get f(t). I got the Laplace transform : $(s^3 - 24s) ...
4
votes
1answer
57 views

Optimal series expansion for “invertability”

Motivation: Often when dealing with physical phenomena, deviations from the model must be considered, so a variable, say $x\in[0,1]$ will be replaced by a power series expansion: $$x'\ \to \ x(1+k ...
0
votes
1answer
64 views

Evaluating a series with some given formula [duplicate]

I have a formula for the power series corresponding to the function $$\frac{z^{3k}}{(3k)!}$$ and I need to evaluate a new series with it but I can't see how to manipulate it even though I've had some ...
2
votes
1answer
93 views

Find the sum of $\sum_{n=1}^\infty \frac{x^{n-1}}{3^nn}$ - What is wrong with my solution?

I have to find the sum of the following power series: NOTE: please assume that x is in the convergence domain. $$\sum_{n=1}^\infty \frac{x^{n-1}}{3^nn}$$ My ...
5
votes
3answers
169 views

Series of inverses of binomial coefficients

Can you think of a simple way of proving that $$ \sum_{n=k+1}^\infty \frac{1}{n \choose k} $$ is rational for any $k \geq 2$? Here's the background. Consider a series: $$ \sum_{n=1}^\infty ...
-1
votes
2answers
63 views

Power series with radius of convergence 2 that diverges at both -2 and 2?

I'm looking for a real power series that has radius of convergence 2 but diverges at both 2 and -2. Any idea? Thank you!
3
votes
1answer
184 views

Finding a radius of convergence of power series

I have to find the radius of convergence of some power series but I find myself in trouble for three of them : the series are $\sum2^kx^{k!}$ $\sum\sinh(k)x^k$ $\sum\sin(k)x^k$. For the first ...
1
vote
2answers
66 views

Finding power series

I need to find the power series for $e^z + e^{az} + e^{a²z}$ where $a$ is the complex number $e^{2πi/3}$. I know that $1 + a + a² = 0$. I have tried to differentiate the expression and give values ...
1
vote
0answers
63 views

Alternative Bound on a Double Geometric Series

If $|a_{mn}x_0^my_0^n| \leq M$ then a double power series $f(x,y) = \sum a_{mn} x^m y^n$ can be 'bounded' by a dominant function of the form $\phi(x,y) = ...
1
vote
1answer
74 views

Ratio of coefficients for Laurent series expansions [duplicate]

Let $f$ be analytic in the disk $D(0,2)$ except for a pole of order $1$ at $z=1$, and let $$f(z)=\sum_{k=0}^\infty a_k z^k$$ be the series expansion for $f$ in the disk $D(0,1)$. Prove that ...
1
vote
2answers
55 views

convergence ratio of the serie $e^{xn}$

How can I determine the values of $x$ such that the series converge: $$\sum_{n=0}^\infty e^{xn}$$ I'm really lost in this problem, please help.
1
vote
2answers
1k views

Converge of the sum $\sum_{k=1}^{n} k x^k $

For what values ​​of x the sum converges and what is the limit when $n \rightarrow \infty$ $\sum_{k=1}^{n} k x^k $ My work: First i try to calculate the interval and radius of convergence of ...
2
votes
3answers
112 views

power series expansion of $z^a$ at $z = 1$

I'm working through some problems in a complex analysis book, and one asks to compute the power series expansion of $f(z) = z^a$ at $z = 1$, where $a$ is a positive real number. The series should ...
2
votes
2answers
106 views

Expansion and convergence of $\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}$

Consider the series: $$\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}\;\;\;\;n\in\mathbb{N}$$ Other than formal manipulation of the Taylor series of the $\sin$ function, is there a way to expand ...
2
votes
2answers
317 views

Given a perturbation of a symmetric matrix, find an expansion for the eigenvalues

Let $A$ be a real, symmetrix $n\times n$ matrix with $n$ distinct, non-zero eigenvalues, and let $V$ be a real, symmetric $n\times n$ matrix. Consider $A_{\varepsilon}=A+\varepsilon V$, a ...
5
votes
4answers
166 views

Compute $\left (1+\frac{1}{2}+\cdots+\frac{1}{n} \right )^2+\cdots+\left (\frac{1}{n-1}+\frac{1}{n} \right )^2+\left (\frac{1}{n} \right )^2$

Compute the value of the following expression $$\left (1+\frac{1}{2}+\cdots+\frac{1}{n} \right )^2+\left ( \frac{1}{2}+\cdots + \frac{1}{n}\right )^2+\cdots+\left (\frac{1}{n-1}+\frac{1}{n} \right ...
2
votes
1answer
88 views

Series $\sum_{n=1}^{+ \infty}\frac{z^{n}}{n}e^{n^{2}z}$

Let $$f(z) = \sum_{n=1}^{+ \infty}\frac{z^{n}}{n}e^{n^{2}z} \ \ \ \ ,z\in \mathbb{C}$$ I want to find the maximal region in which $f$ is holomorphic. I have a problem with the convergence in $\{+i, -i ...
2
votes
1answer
212 views

Formal series expansion of differential operator (d/dx + f(x))^n

My original problem was to find the "coefficient" functions $\varphi_{k,n}(x)$ in $$ (\partial_x + f(x))^np(x) = \sum_{k=0}^n\varphi_{k,n}(x)\partial_x^kp(x). $$ (i.e. find the coefficients ...
1
vote
1answer
127 views

Find the radius of convergence for $\sum^{\infty}_0 n^nz^{n^n}$

Find the radius of convergence for $\sum^{\infty}_{n=0} n^nz^{n^n}$ This is not a power series, but if I define $a_k=k$ if $k=n^n$ and $a_k=0$ otherwise, I would have a power series such that ...
2
votes
1answer
324 views

Reversion of power series

So, I just heard about this method. How does one determine the coefficients, and what is it used for? For example, given $$ y = x - \frac{x^3}{6} + \frac{x^5}{120} + O(x^7)$$ reversion would give a ...
1
vote
2answers
117 views

Is there a real power series with radius of convergence 1 that converges at 1 but not at -1?

I can find a power series that has radius of convergence 1 but since any series that converges absolutely converges, I cannot find any that converges at 1 but diverges at -1... Can you help me? Thank ...
3
votes
1answer
245 views

Convergence radius of $\sqrt{\cos(z)}$

Compute the first 3 non zero terms of the Taylor expansion of $\sqrt{\cos(z)}$ at $z=0$ and determine its convergence radius, considering only the principal branch of the square root. I've computed ...
3
votes
1answer
58 views

Analytic function satisfying $x^2f''(x) +xf'(x)+x^2f(x)=0$ and $f(0)=1$

Suppose that the power series $\sum_{n=0}^\infty a_nx^n$ converges for all real $x$ to a function $f(x)$ that satisfies $$x^2f''(x) +xf'(x)+x^2f(x)=0 \quad \text{and} \quad f(0)=1.$$ ...
2
votes
2answers
89 views

Formal power series with all derivatives zero

I have the following question. Suppose I have a formal power series $f(x)=\sum\limits_{i=0}^\infty c_ix^i$ with real coefficients. Suppose that all the derivatives ...
1
vote
1answer
110 views

Convergence radius: $R = \lim_{n \rightarrow \infty} \frac {\mid a_n \mid} {\mid a_{n+1} \mid}$ (incl. $\infty$) when $R = 0$ and Ratio test

I have read the following proof of a theorem in a textbook of mine, and I've been wondering why the proof holds when $$R = \lim_{n \rightarrow \infty} \frac {\mid a_n \mid} {\mid a_{n+1} \mid} = 0$$ ...
23
votes
1answer
765 views

Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius?

It is well-known that the function $$f(x) = \begin{cases} e^{-1/x^2}, \mbox{if } x \ne 0 \\ 0, \mbox{if } x = 0\end{cases}$$ is smooth everywhere, yet not analytic at $x = 0$. In particular, its ...
6
votes
1answer
111 views

Simplify $\sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x)$

Simplify the following expression $$S_N = \sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x), $$ where $a$ is a real number and $f(x)$ is an analytic real function. What is $\lim_n ...
0
votes
2answers
142 views

Maclaurin Series confusion

Using the Maclaurin expansion formula: to find the Maclaurin series for $sin(3x)$, I can get the correct answer by using $x^n$ in the formula above (in the tail-end of the formula). Similarly, to ...
3
votes
2answers
130 views

How to find the radius of convergence?

The function is $\dfrac {z-z^3}{\sin {\pi z}} $. How to find the radius of convergence in $ z=0 $?
3
votes
1answer
116 views

Finding complex power series with interesting boundary behavior

I need to find one (or more) interesting complex power series to give to my students for their analysis exam. Ideally, this would be a power series that has interesting behavior at the boundary, i.e. ...
12
votes
2answers
347 views

Finding the derivative of $x\uparrow\uparrow n$

I am trying to find a general derivative for the function: $f(x)=x^{x^{x^{...^{x}}}}$however to do that I must find $f^{\prime }$ and $f^{\prime \prime}$...etc. I am now trying to write down a general ...
1
vote
2answers
58 views

An easy question on complex

Let $\{u_{k}\}_{k=1}^{\infty}$ be a complex number sequence. If $\sum_{k=1}^{\infty}\lambda^{k}u_{k}=0$, for each $\lambda\in \mathbb{D}(0, 1/3)$(where the $\mathbb{D}(0, 1/3)~$denotes an open disc ...
1
vote
1answer
104 views

Prove $\sum_{m \geq 1} {\frac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \frac{t^m}{(1-x)^{2m-1}}} $is divergent

How do I show that the following power series is divergent? $$ u(t,x) = \sum_{m \geq 1} {\dfrac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \dfrac{t^m}{(1-x)^{2m-1}}} $$ where $t$ is complex 1-dimensional, $x$ ...
4
votes
2answers
399 views

Maclaurin expansion of arctan: convergence?

In my textbook, the Maclaurin series expansion of $\arctan{x}$ is found by integrating a geometric series, that is, by noting that $\frac{d}{dx}(\arctan(x)) = \frac{1}{x^2+1}$ then rewriting the ...