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Homework Statement
Let {a_n} be positive, decreasing. Show that if a_1 + a_2 + a_3 + ... converges then lim n * a_n = 0.
Homework Equations
None.
The Attempt at a Solution
Consider the harmonic series 1 + 1/2 + 1/3 + ... . Observe that
[a_n] / [1 / n] = n * a_n .
Since 1 + 1/2 + 1/3 + ... diverges, by the Limit Comparison Test we know that, if it exists, lim n * a_n = 0, for otherwise a_1 + a_2 + a_3 + ... would also diverge, a contradiction.
My question is, how do I prove that the limit necessarily exists? Considering f(x) = 1/x, I imagine, for example, a function g(x) in the shape of a step function such that liminf g(x) / f(x) = 0 and limsup g(x) / f(x) = 1. In this case the limit would not exist so I guess (in light of the problem statement) that the series g(1) + g(2) + g(3) + ... must diverge, but I don't know how to show this.
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