Which of the following doesnot ensure the convergence of real sequence ${a_n}$
A. |$a_n - a_{n+1}|$ goes to zero as n goes to infinity
B .$\sum\limits_{n=1}^\infty$ |$a_n - a_{n+1}|$ is convergent
C. $\sum\limits_{n=1}^\infty$ $na_n$ is convergent
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@user184794 sorry it was a typo .i have edited the q – Sophie Clad Nov 18 '14 at 07:35
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Please add your thoughts on the problem for better answers – Learnmore Nov 18 '14 at 07:46
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@learnmore I was taking a convergent sequence (also took harmonic numbers ) and was checking with conditions so that after doing calculations it becomes divergent , which ain't happened . But here answer is just the reverse of what i was doing – Sophie Clad Nov 18 '14 at 07:48
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In order to show that a condition does not ensure the convergence of a sequence, we must show there exists a sequence which fulfills the condition, but is not convergent.
The sequence of harmonic numbers satisfies condition A:
$$H_n = \sum_{k=1}^{n}\frac 1 k\\ \left|H_n-H_{n+1}\right| = \left|-\frac 1 {n+1}\right| = \frac 1 {n+1}\to0$$
But it is not convergent (shown here or on wikipedia).
So condition A does not ensure convergence.
