Problem 1126 (difficulty: 5/10)

Let \(\displaystyle \sum\limits_{n=1}^\infty a_n\) has positive terms.

(a) Prove that if \(\displaystyle \displaystyle \liminf~ n\left(\frac{a_n}{a_{n+1}}-1\right)>1\), then the series is convergent.

(b) Prove that if \(\displaystyle \displaystyle n\left(\frac{a_n}{a_{n+1}}-1\right)\le1\) for \(\displaystyle n\) large enough, then the series is divergent.

(Raabe-criterion)

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