Harmonic series diverges and pseudo Cauchy however it's not bounded. So how can I find such a sequence?
A sequence $(s_n)$ is pseudo-Cauchy if, for all $\xi>0$, there exists an $N$ such that if $n ≥ N$, then $|s_{n+1}−s_n| < ξ$.
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5What is a pseudo-Cauchy sequence? – Oct 16 '13 at 17:39
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3$$\textstyle 0,,{1\over 2},, 1,, {2\over 3},,{1\over 3},,{0},,{1\over 4},,{2\over 4},,{3\over 4},,1,,{4\over 5},,{3\over 5},,{2\over 5},,{1\over 5}, ,0, \ldots$$ – David Mitra Oct 16 '13 at 17:42
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1A sequence (sn) is pseudo-Cauchy if, for all ξ> 0, there exists an N such that if n ≥ N, then |sn+1−sn| < ξ – user11111 Oct 16 '13 at 17:42
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@DavidMitra How would you go about proving the sequence using the definition? I'm struggling to understand how to begin that part. – DoubleV Oct 09 '22 at 23:07
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Try the sequence $$a_n=\sin\sqrt n$$
Hagen von Eitzen
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@user11111 $\sqrt{n+1}-\sqrt n\rightarrow 0$ as $n\rightarrow\infty$. Nicer than my example. – David Mitra Oct 16 '13 at 18:01
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@user11111 My last comment implies Hagen's sequence is indeed pseudo-Cauchy. – David Mitra Oct 16 '13 at 18:06
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Be careful here, we want to show |sn+1−sn| < ξ but not n+1 - n .So Sin(n) can be any number when n goes to infinite, thus it's not pseudo Cauchy. – user11111 Oct 16 '13 at 18:14
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2@user11111 By the Mean Value Theorem, $|\sin\sqrt{n+1}-\sin\sqrt n|\le|\sqrt{n+1}- \sqrt n|{\rightarrow} 0$. – David Mitra Oct 16 '13 at 18:18
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Mean value Theorem is applied to a continuous function, can we use it in a discontinuous Sequence? – user11111 Oct 16 '13 at 18:23
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I'm still not sure about this one. We are talking about the sequence sin(1), sin(2), sin(3)..... but not the sin function. – user11111 Oct 16 '13 at 18:28