Recurrence relations, convergence tests, identifying sequences
15
votes
0answers
635 views
Prove that sum is finite
Let $j \in \mathbb{N}$. Set
$$
a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!}
$$
and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$.
Please help me to prove that the following sum is ...
12
votes
0answers
192 views
Binomial sum of $n$ terms in closed form
Can we calculate the given combinatorial sum in closed form?
$$ \frac{\binom{2}{0}}{1}+\frac{\binom{4}{1}}{2}+\frac{\binom{8}{2}}{3}+\frac{\binom{16}{3}}{4}+\cdots+\frac{\binom{2^n}{n-1}}{n}$$
10
votes
0answers
91 views
Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes
Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes.
I've been thinking about this question, but ...
10
votes
0answers
157 views
Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality
By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent.
Is is true that ...
9
votes
0answers
132 views
Yet another nested radical
Consider $$F(x) = \sqrt{x -\sqrt{2x - \sqrt{3x - \cdots}}}$$
I believe I can prove (with some handwaving) that
$F$ does converge everywhere in C
$\Im F = 0$ for sufficiently large real x (actually ...
9
votes
0answers
161 views
When is an infinite product of natural numbers regularizable?
I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like
$$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$
and
$$\infty ...
8
votes
0answers
63 views
Ramanujan style nested differential Equation
So I was exploring some math the other day... and I came across the following neat identity:
Given $y$ is a function of $x$ ($y(x)$) and
$$
y = 1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + ...
8
votes
0answers
183 views
+50
Asymptotic related to the infinite product of sine
The amount is somewhat complicated ($x$ is a constant):
$$S_n=\sum_{k=1}^n\ln\left(1-\frac{\sin^2\big(x/(2n+1)\big)}{\sin^2\big(k\pi/(2n+1)\big)}\right)\tag{*}$$
I want to enrich my handy powerful ...
8
votes
0answers
329 views
Quadratic Recurrence Relation
The following sequence appeared in IMC 2012 (a math competition): $a_1 = \frac{1}{2}, a_{n+1} = \frac{n a_n^2}{1+(n+1)a_n}$.
I am trying to find an explicit formula for the sequence. It seems to be ...
7
votes
0answers
86 views
All nested partial sums of a sequence tend to $0$. Is the sequence constant?
$S^0:\mathbb N\to \Bbb R$ is a function. For any $m\in \Bbb N$, we define
$$S^m:\Bbb N\to \Bbb R$$
$$S^m(n)=\sum_{k=1}^n S^{m-1}(k)$$
For each $m\in \Bbb N$, we have:
$$\lim_{n \to \infty}S^m(n)=0$$
...
7
votes
0answers
112 views
Consider the sequence defined: $a_1=0, a_{n+1}=3+\sqrt{11+a_n}$, show that is bounded above and increasing using induction.
Consider the sequence defined: $a_1=0, a_{n+1}=3+\sqrt{11+a_n}$
a) Show, using induction, that this sequence is bounded above by 14; b) prove that the sequence is increasing; c) Why must it converge?; ...
7
votes
0answers
112 views
How to prove that my sum coincides with a sequence on the Online Encyclopedia of Integer Sequences?
I have a topological space $X$ whose reduced $\bmod 2$ Betti numbers (that is to say, the dimension of the $\bmod 2$ reduced homology) I computed to be
$$\small
\dim \tilde{H}_t(X; {\mathbb{Z}}_2) = ...
7
votes
0answers
154 views
Can all subseries of an infinite series be pairwise independent over $\mathbb{Q}$?
I'm wondering about a simple question that has multiple possible variants depending on a few parameters. The prototypical one would be:
Does there exist an infinite series such that any two ...
7
votes
0answers
169 views
Invariant functions on the space of finite sequences of reals
Let $S$ be a space of all finite sequences of real numbers (we don't endow it with metric or topology in general). Before asking the main question, some notation.
For each $\mathbf s\in S$ we define ...
6
votes
0answers
22 views
Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$
This question came up in the process of finding solution to another problem. Eventually, the problem was solved avoiding calculation of this sum, but it looks quite interesting on its own. Is there a ...
