# What can we say If lim U(n+1) - U(n) = 0?

So i proved that $$\lim_{n\to \infty} U(n+1) - U(n) = 0$$ and we also have $a< U(n) < b$ What can we say?

Does this prove that the serie converge?

• Is $U(n)$ a sequence of numbers?
– user99914
Nov 7 '14 at 22:33

Counter-example: $\sin(\sum_{i=1}^{n}{\frac{1}{i}})$.
Consider the series $$1-\frac{1}{2}-\frac{1}{2}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}- \frac{1}{6}-\frac{1}{6}-\cdots.$$