# A sequence $\{a_n\}$ that diverges but $\displaystyle\lim_{n\to 0} |a_n-a_{n+1}| =0$ [duplicate]

What is a sequence $\{a_n\}$ that diverges and $\displaystyle\lim_{n\to 0} |a_n-a_{n+1}| =0$

• Do you mean $a_{n+1}$ or $a_n + 1$? Btw I think you want $n \to \infty$. – MathematicsStudent1122 Dec 11 '16 at 8:40
• Do you want the sequence to diverge, or the sum to diverge. Because $a_n = 1$ converges perfectly fine $a_n\to 1$, but $\sum_n a_n$ diverges. – Mark Dec 11 '16 at 8:40
• – Martin Sleziak Dec 11 '16 at 11:50
• By the way, if $\lvert a_{n+1} - a_n \rvert \leq u_n$, and $\sum_{n=1}^{\infty}u_n$ converges, then $(a_n)$ is Cauchy. – Desura Dec 11 '16 at 12:26

Take $$a_n=\sqrt{n}$$ then note that, for $n\ge1$, $$a_{n+1}-a_n=\sqrt{n+1}-\sqrt{n}=\frac1{\sqrt{n+1}+\sqrt{n}}.$$
$$a_n:=\log n\implies a_{n+1}-a_n=\log\left(1+\frac1n\right)\xrightarrow[n\to\infty]{}\;\ldots$$