Recurrence relations, convergence tests, identifying sequences
21
votes
3answers
313 views
+500
Conjectural closed-form representations of sums, products or integrals
What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...
12
votes
2answers
244 views
+50
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:
$$\lim_{n \to \infty}S^m(n)=0$$
...
13
votes
1answer
138 views
+50
Inequality in a bounded real sequence
Prove or disprove that for any bounded real sequence $\{x_n\}_{n\in\mathbb{N}}$ there exist two distinct natural numbers $u,v$ such that:
$$|x_u-x_v|\cdot|u-v|\leq 1.$$
