<s>[INST] 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?
 [/INST] Consider the series
$$1-\frac{1}{2}-\frac{1}{2}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}- \frac{1}{6}-\frac{1}{6}-\cdots.$$
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