What is a sequence $\{a_n\}$ that diverges and $\displaystyle\lim_{n\to 0} |a_n-a_{n+1}| =0$
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Do you mean $a_{n+1}$ or $a_n + 1$? Btw I think you want $n \to \infty$. – MathematicsStudent1122 Dec 11 '16 at 08:40
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2Do 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 Schultz-Wu Dec 11 '16 at 08:40
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See also: Pseudo-Cauchy sequence, I want an example of a sequence that satisfies $|x(n) - x(n-1)| \to 0$ but not Cauchy, If ${x_n}$ satisfies that $x_{n+1} - x_n$ goes to $0$, is ${x_n}$ a Cauchy sequence?, Cauchy Sequence of Real Numbers, etc. – Martin Sleziak Dec 11 '16 at 11:50
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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
3 Answers
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Take $$ a_n=\sqrt{n} $$ then note that, for $n\ge1$, $$ a_{n+1}-a_n=\sqrt{n+1}-\sqrt{n}=\frac1{\sqrt{n+1}+\sqrt{n}}. $$
Olivier Oloa
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Hint: consider harmonic series.
jnyan
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This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review – Björn Friedrich Dec 11 '16 at 09:13
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3@BjörnFriedrich "This does not provide an answer to the question." Actually it does, very much so. – Did Dec 11 '16 at 09:26
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@BjörnFriedrich It very much hints to an answer of the question. Perhaps adding the words 'the partial sums sequence of" between "consider" and harmonic" makes this more direct, but hints are very welcome. – DonAntonio Dec 11 '16 at 09:57
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Besides the two good answers you already got, you could also take
$$a_n:=\log n\implies a_{n+1}-a_n=\log\left(1+\frac1n\right)\xrightarrow[n\to\infty]{}\;\ldots$$
DonAntonio
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