I am looking for an example of a sequence which shows that, $a_{n+1} - a_n \to 0$ as $n \to \infty$ does not imply that sequence $a_n$ converges. I have a feeling that the sequence should be an oscillatory one but I am unable to think of an example.
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1Consider the sequence of partial sums of a nonconvergent series. – Oct 27 '15 at 16:09
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2Asked tons of times on the site... Try $a_n=\log n$. – Did Oct 27 '15 at 16:14
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The most elementary example is probably $a_n=\sqrt{n}$. It is relatively easy to show that $\sqrt{1+n}-\sqrt{n}\to 0$. – Thomas Andrews Oct 27 '15 at 16:17
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$0$, $1/2$, $1$, $3/4$, $2/4$, $1/4$, $0$, $1/8$, $\ldots$. – David Mitra Oct 27 '15 at 16:21
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Also https://math.stackexchange.com/questions/1437381/bounded-sequence-which-is-not-convergent-but-differences-of-consecutive-terms-c which is perhaps a better target. – Eric Wofsey Feb 28 '20 at 21:39
