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).
39
votes
1answer
652 views
Fractal behavior along the boundary of convergence?
The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, ...
30
votes
3answers
1k views
Factorial and exponential dual identities
There are two identities that have a seemingly dual correspondence:
$$e^x = \sum_{n\ge0} {x^n\over n!}$$
and
$$n! = \int_0^{\infty} {x^n\over e^x}\ dx.$$
Is there anything to this comparison? (I ...
23
votes
2answers
630 views
On calculating $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ — where is the mistake?
I've seen the integral $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ on a thread in this forum and I tried to calculate it by using power series. I wrote the integral as a sum then again as an integral. ...
21
votes
1answer
633 views
How to prove convergence of polynomials in $e$ (Euler's number)
These polynomials in $e$ converge to 2$$f(i)=e^i - i \sum_{k=1}^{i-1}\frac{(i-k)^{k-1}{e^{i-k}}{(-1)^{k+1}}}{k!}, \text{ where } i>1$$
This function goes to 2. I've calculated this with sage math ...
20
votes
2answers
992 views
Proving an “amazing” claim regarding $\zeta( 3)$ and Apéry's proof
I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$
$$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$
and thus (probably) ...
19
votes
5answers
526 views
Power Series Summation Formula
I am wondering if there exists any formula for the following power series :
$$S = r + r^2 + r^4 + r^8 + r^{16} + r^{32} + ...... + r^{2^k}$$
Is there any way to calculate the sum of above series (if ...
16
votes
3answers
367 views
Math contest: Find number of roots of $F(x)=\frac{n}{2}$ involving a strange integral.
Edit summary: A good answer appeared. CW full answer added, based on given answers. Removing my ugly-looking attempts, as they still remain in the rev. history.
Here's a final-round calculus ...
15
votes
3answers
451 views
Multiplicative inverses of formal series with non-negative coefficients
What are the formal series $f$ with non-negative integer coefficients and constant term equal to $1$ whose multiplicative inverse $1/f$ has all coefficients, apart from a finite subset, all ...
14
votes
1answer
222 views
Necessary and sufficient conditions for a polynomial in $\mathbb{Z}[t]$ to have an $n$th root in $\mathbb{Z}[[t]]$
Let $p(t) = \sum p_k t^k$ be a polynomial in $\mathbb{Z}[t]$, with $p_0=1$. Is there a necessary and sufficient condition (congruence or other) on the coefficients $p_k$ such that $p(t)$ admits a ...
13
votes
3answers
395 views
Solving a difficult recursion via generating functions
I have been trying to solve the recurrence:
\begin{align*}
a_{n+1}=\frac{2(n+1)a_n+5((n+1)!)}{3},
\end{align*}
where $a_0=5$, via generating functions with little success. My progress until now is ...
12
votes
2answers
298 views
Finding the convergence interval of $\sum\limits_{n=0}^{\infty} \frac{n!x^n}{n^n}$.
I want to find the convergence interval of the infinite series $\sum\limits_{n=0}^{\infty} \dfrac{n!x^n}{n^n}$.
I will use the ratio test: if I call $u_n = \dfrac{n!x^n}{n^n}$, the ratio test says ...
12
votes
1answer
394 views
Does a power series vanish on the circle of convergence imply that the power series equals to zero?
Let $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be a power series, $a_n, z\in \mathbb{C}$. Suppose the radius of convergence of $f$ is $1$, and $f$ is convergent at every point of the unit circle.
Question:If ...
12
votes
2answers
354 views
$\lim\limits_{x\to\infty}f(x)^{1/x}$ where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$.
Does the following limit exist? What is the value of it if it exists?
$$\lim\limits_{x\to\infty}f(x)^{1/x}$$
where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$ and $\{a_k\}\subset\mathbb{N}$ ...
11
votes
4answers
663 views
How to express $(1+x+x^2+\cdots+x^m)^n$ as a power series?
Is it possible to express $(1+x+x^2+\cdots+x^m)^n$ as a power series?
10
votes
4answers
785 views
Help with summing a power series
I'd like to determine the function corresponding to the following power series:
$$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1},
$$
where ...
10
votes
4answers
307 views
Finding the power series for $y$ where $y + \sin(y) = x$
What do you do to find the power series for an inverse relationship such as for $y$ in $y + \sin(y) = x$? I'm not sure where to begin.
(Similarly, the Lambert $W$ function has such a power series ...
10
votes
4answers
309 views
Detecting a negative coefficient in a power series
Suppose that I have an analytic function $f(z)=\sum_{n=0}^\infty a_n z^n$ which converges on some disk around the origin.
For a particular function I encountered, I wished to prove that every ...
10
votes
2answers
195 views
How to do a very long division: continued fraction for tan
I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
10
votes
1answer
387 views
Radius of convergence of power series
Given a meromorphic function on $\mathbb{C}$, is the radius of convergence in a regular point exactly the distance to the closest pole?
As Robert Israel points out in his answer, that this is of ...
10
votes
3answers
602 views
Product of two power series
Say if I define a power series over some arbitrary field $F$ as
$$a = \sum^{ \infty }_{i = 0} a_{i} X^{i} $$
Then can I say:
$$ab = \sum^{ \infty }_{i = 0} \sum^{ \infty }_{j = 0} a_{i} b_{j} X^{i ...
10
votes
1answer
164 views
Abel limit theorem
I would like to know if the Abel limit theorem works if the limit is infinite.
Let the series $\sum_{k=0}^\infty a_k x^k$ have radius of convergence 1. Assume further that $\sum_{k=0}^\infty a_k = ...
9
votes
3answers
154 views
Can the the radius of convergence increase due to composition of two power series?
When composing power series, is the radius of convergence the minimum of that of the individual series, or is it like for multiplication and addition of power series where the resultant radius of ...
9
votes
1answer
97 views
Is this (classical?) exercice missing a hypothesis?
A friend just told me about an exercice he was given quite a few years ago, but he wasn't sure wether he remembered all the hypothesis correctly. Does anybody recognize this?
Let $f$ be a smooth ...
9
votes
0answers
498 views
any pattern here ? (revised)
for any positive number $k$, I have a $(k+1)*(k+1)$ matrix. I wonder if these matrices follow any "obvious" pattern. My goal is to guess the elements for matrix with $k=5$ and above (most probably in ...
7
votes
3answers
428 views
The Power of Taylor Series
I am teaching a Calculus class and we are finishing up power/Taylor series this week. The last section of the chapter is on applications, but the only ones listed there are approximating non-rational ...
7
votes
2answers
267 views
Summing Matrix Series
I need to sum the series
$$I + A + A^2 + \ldots$$
for the matrix
$$A = \left(\begin{array}{rr}
0 & \epsilon \\
-\epsilon & 0
\end{array}\right)$$
and $\epsilon$ small. The goal is to ...
7
votes
5answers
340 views
Formal power series - a question
I've been reading generationgfunctionology by Herbert S. Wilf (you can find a copy of the second edition on the author's page here).
On page 33 he does something I find weird. He wants to shuffle the ...
7
votes
3answers
455 views
How to calculate $f(x)$ in $f(f(x)) = e^x$?
How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$? Is there a faster-converging method than power series for fractional iteration/functional square roots?
7
votes
3answers
151 views
Calculating value of $1000^{th}$ derivative at $0$.
I need to calculate value of $1000^{th}$ derivate of the following function at $0$:
$$
f(x) = \frac{x+1}{(x-1)(x-2)}
$$
I've done similar problems before (e.g. $f(x)= \dfrac{x}{e^{x}}$) but the ...
7
votes
2answers
285 views
Power series without analytic continuation
Given a formal power series $\sum a_n z^n$ and a radius of convergence $R>0$, there are various ways to extend the function to the boundary such as
Abel's theorem
Fatou's lemma
$H^\infty$ ...
7
votes
5answers
239 views
When are we (not) allowed to replace $x$ by $ix$?
It seems to be quite a common manipulation to replace $x$ by $ix$.
Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
7
votes
2answers
201 views
Proving that $\pi=\sum\limits_{k=0}^{\infty}(-1)^{k}\left(\frac{2^{2k+1}+(-1)^{k}}{(4k+1)2^{4k}}+ \frac{2^{2k+2}+(-1)^{k+1}}{(4k+3)2^{4k+2}}\right)$
Long time ago I've been playng with formulas for $\pi$ and found that one above in the title which can also be expressed as
\begin{align*}
...
7
votes
1answer
107 views
Does a convergent power series on a closed disk always converge uniformly?
If I have a power series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i} \in\mathbb{C}[[z]]$ with radius of convergence $r>0$ and I know that the series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
7
votes
1answer
188 views
Infinite series where each term is the square of the last
Is there a closed-form, in terms of elementary functions or otherwise, for the power series $x+x^2+x^4+x^8+x^{16}+...$, where each term is the square of the last?
7
votes
1answer
297 views
The power series $\sum\limits_{n=1}^{\infty} \frac {z^{n} }{ n^{2}} \ $
This is an exercise from Remmert.
The power series $\sum\limits_{n=1}^{\infty} \frac {z^{n} }{ n^{2}} \ $ has radius of convergence $1 \ $. Show that the function it represents is injective in $\{ z ...
7
votes
1answer
226 views
Solving a formal power series equation
I want to find a function $f(x,y)$ which can satisfy the following equation,
$$\prod _{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = \exp \left[ \sum _{n=1} ^\infty ...
6
votes
4answers
323 views
Power Series with the coefficients $n!/(n^n)$
I'd be grateful if someone could tell me how to obtain the convergence radius of the aforementioned power series.
Or, by Cauchy Hadamard, the limit of
$(n!/(n^n))^{(1/n)}$ as n approaches infinity.
...
6
votes
4answers
839 views
Formal power series coefficient multiplication
Given that I have two formal power series:
$$ A(x) = \sum_{k \ge 0} a_k x^k $$
$$ B(x) = \sum_{k \ge 0} b_k x^k $$
The Cauchy Product gives a series
$$ C(x) = \sum_{k \ge 0} c_k x^k $$
$$ c_k = ...
6
votes
3answers
218 views
Expressing the area of the image of a holomorphic function by the coefficients of its expansion
I have the following problem.
Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set ...
6
votes
2answers
400 views
What is the radius of convergence of $\displaystyle\sum z^{n!}$?
How could you find out the radius of convergence of $\displaystyle\sum z^{n!}$? I'm used to applying the ratio test to power series of the form $\displaystyle\sum a_{n}z^{n}$, but for a different ...
6
votes
2answers
166 views
Power series $x f''(x) + f'(x) + xf(x) = 0$
Find a power series with radius of convergence $R = \infty$ such that $f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$ satisfies $x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R$.
How should ...
6
votes
2answers
130 views
Ring of formal power series finitely generated as algebra?
I'm asked if the ring of formal power series is finitely generated as a $K$-algebra. Intuition says no, but I don't know where to start. Any hint or suggestion?
6
votes
3answers
228 views
When can we plug an arbitrary number into an equation on formal power series
We often ignore convergence when manipulating formal power series. We can add, substract, multiply, divide, differentiate, and do functional composition on such formal sums without worrying about ...
6
votes
4answers
353 views
what is the fraction field of $R[[x]]$, the power series over some ring?
I have a question similar to 74335.
Let $R$ be an integral domain. Is there a nice description of the fraction field of the power series $R[[x]]$?
I know that this field can be a proper subfield of ...
6
votes
1answer
139 views
Power Series Expansion
I was flicking through a book on perturbation methods and saw a simple question asking the reader to expand the following expression for $f$ in a power series (up to the first 2 terms):
$f = (1 + ...
6
votes
1answer
157 views
sum of infinite series
Does there exist an explicit expression for
$$\sum_{k=0}^{\infty }\frac{\left( -1\right) ^{k}e^{-\lambda \left(
2k+1\right) ^{2}}}{\left( 2k+1\right) ^{3}}\;,$$
where $\lambda$ is a positive scalar? ...
6
votes
1answer
72 views
Existence of a power series converging non-uniformly to a continuous function
I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that:
$f$ converges and is continuous on the closed unit disk $D$ and
the series $\sum_n a_n z^n$ does not converge ...
6
votes
1answer
96 views
Prime elements in $\mathbb{Q}[[X,Y,Z]]$ whose status as an infinite series is unchanged by arbitrary multiplication
Let's suppose $R$ is the ring $\mathbb{Q}[[X,Y,Z]]$. I'm interested in finding power series $f(x,y,z) \in R \setminus \mathbb{Q}[X,Y,Z]$ which are, first of all, prime elements in $R$, but also ...
6
votes
1answer
178 views
Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$
The following is a historical question, but first some background:
Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be ...
6
votes
1answer
68 views
A question regarding representation of a function as a power series
I'm trying to help my brother with a calculus problem related to the representation of a function as a power series. The task is to find what power series is represented by the following function, and ...
