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Let $(u_n)$ be a non increasing sequence of reals such that $\sum u_n$ converges.

I'm investigating the question: do we always have $u_n = o(1/n)$? I suspect not and I'm trying to build a sequence $(u_k)$ defined by $u_k = 1/\varphi_{n+1}$ for $k \in (\varphi_n, \varphi_{n+1}]$ where $(\varphi_n)$ is an increasing sequence of natural numbers. For the time being, I'm not succeeding in my construction...

Do you think that it is possible to build a sequence based on the construction above and satisfying the required assumptions? If yes can you provide such an example?

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