Does a Sequence Converge if Its Differences Tend to Zero?

[ppy]

Hi,

Let a(n) be a real sequence such that a(n+1)-a(n) tends to zero as n approaches ∞. must a(n) converge? Also an explanation would be great thank you. have been wondering about this

 

[HallsofIvy]

No.

Let $$a_n= \sum_{i=1}^n \frac{1}{i}$$.

Then $a_{n+1}- a_n= 1/(n+1)$ which goes to 0 as n goes to infinity. But the harmonic series does NOT converge so this sequence does not converge.