All Questions
13 questions
7
votes
1
answer
348
views
Descartes' rule of signs for infinite series
Consider the function given by
$$f(x)=1-a_1x-a_2x^2-a_3x^3-\cdots$$
where each $a_k\geq0$ and some $a_j>0$. If $f(x)$ is a polynomial then Descartes' Rule of signs tells us there is exactly one ...
- 43.2k
6
votes
1
answer
351
views
Looking for infinite series resembling an exponential
I'm looking for some $f(x)$ that has the following property:
$\sum_{x=1}^\infty f(kx) = r^k$
for some real $0 < r < 1$, and at least for strictly positive integer $k$.
Does such an $f(x)$ ...
- 4,897
6
votes
3
answers
626
views
Alternating power series $\sum_{k\geq 0}(-1)^k z^{(2k+1)^2}$
Suppose that $f(x):=\sum_{k\geq 0}(-1)^ke^{-(2k+1)^2x}$ has a holomorphic continuation to a neighborhood of $0$, that is, $f(x)=\sum_{n\geq 0}a_n x^n$ for $x> 0$ small. I want to know the value of $...
- 61
5
votes
2
answers
874
views
Searching for a proof for a series identity
The below identity I have found experimentally.
Question. Is this true? If so, may you provide a "slick" (or any) proof.
$$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...
- 43.2k
4
votes
1
answer
155
views
Could the range of $\sum_{k\geq 1}r^{n(k)}$ for $r\in \big(\frac{1}{2}, 1\big)$ be continuous?
Let $\mathcal{F}_N$ be the set of all strictly increasing sequences of positive integers. For every two $F_1, F_2\in\mathcal{F}_N$, if we use $\delta(F_1,F_2)$ to denote the first $n$-th coordinate ...
- 307
4
votes
0
answers
459
views
Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?
Is there any closed form expression for the following serie?
$$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$
Or at least a proof that it is an irrational number. The ...
3
votes
1
answer
379
views
Convergence of a power series
Consider the numbers $$a_n=\frac{1}{n+1}\sum_{k=0}^{n}\frac{2^{k-1}\binom{n+1}{k}B_k}{2^{s+k-1}-1}, \ n\geq0,$$ where $s\neq1;0;-1;-2;-3;...$ is a fixed real number, and the $B_k$ are the Bernoulli ...
- 463
2
votes
1
answer
210
views
asymptotic estimate for log-tan sum
I am finding the following first order estimate.
Question. As $y\rightarrow\infty$,
$$\sum_{n=1}^{\infty}\frac{\log n}n\,\arctan\frac{y}n\,\,
\sim\,\,\frac{\pi}4\log^2y.$$
Is it true?
- 43.2k
2
votes
2
answers
783
views
Weak version of Karamata's Tauberian theorem
I first posted this on mathematics. However, I got no answer there and it seems adapted here too. Also, it seems to be harder than I first thought.
Karamata's Tauberian theorem states the following. ...
- 2,090
1
vote
2
answers
346
views
Is this relationship, $\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$, true?
According to numerical simulation, the relationship
$$\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$$
where $\Gamma$ is the Gamma function seems to be true.
Do you ...
user365311
1
vote
1
answer
111
views
Can we write $e^{-\alpha x}$ as $\sum_{n=0}^\infty c_n\left(\alpha\right)\gamma\left(x\right)^n$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$
Do there exist continuous functions $c_n\colon\mathbb R^+\to\mathbb R$ and $\gamma\colon\mathbb R^+\to\mathbb R$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$ and the following equation is true ...
- 438
1
vote
0
answers
51
views
Convergence acceleration of a series by using optimal parameters
One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a ...
- 5,470
0
votes
0
answers
111
views
Sum power series not continuous unit circle
This is (probably) not a research question and I already asked it on StackExchange but I got no answer over there.
Let us consider the sequence $(a_n)_{\geq 1} = \left(\frac{\cos(2\sqrt{2n}+\frac{\pi}{...
- 7,300