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While studying Leibniz test for infinite alternating series I am having an intuition, consider a series $$ \sum_{n=1}^{\infty}(-1)^{n+1}u_{n} $$ If $$ \left \{ u_{n} \right \} $$ is a non-decreasing sequence of non-negative real numbers, such that $$ \lim_{n\rightarrow \infty} u_{n} = 0 $$ Then, the series does not converge.
Can we prove this? or can anyone please provide a counter example to it?
1 Answer
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If $u_n$ is non-decreasing, non-negative and $ \lim u_n=0$ then $u_n=0$ for all $n$ so the series converges.
If 'non-decreasing' is changed to 'non-increasing' then convergence follows by Leibniz test.
