TITLE: Limit of sequence of fractions
QUESTION [2 upvotes]: Given sequence $a_n = (\frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{(n+1)^2})$. How to prove that $\lim_{n \to \infty}{a_n} = 0$ and $a_n$ is not increasing?

REPLY [0 votes]: The simple lower bounding
$$
\frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{(n+1)^2}\ge
\frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n}\ge\frac{n}{2n}=\frac12
$$
proves that $a_n\not\to 0$.
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