Answer to Question #303202 in Real Analysis for Nikhil

Question

Check whether the sequence {an}, where

an= 1/(n+1)+1/(n+2)+..1/2n is convergent or not

Answer 1

ANSWER. The sequence $\left \{ a_{n} \right \}$ is convergent.

EXPLANATION.

Since $a_{n}=\sum_{k=1}^{n}\frac{1}{n+k}$ , then $a_{n+1}=\sum_{k=1}^{n+1}\frac{1}{(n+1) +k}$ and $a_{n+1} -a_{n}=\frac{1}{2n+1}+\frac{1}{2n+2}-\frac{1}{n+1}=\frac{1}{2n+1}-\frac{1}{2n+2} =\frac{1}{(2n+1)(2n+2)}>0$ . Hence $a_{n+1}>a_{n }$ . So , the sequence is increasing.

Because for all $k$ such that $1\leq k\leq n$ the inequality $(1+n)\leq(k+n)$ is true and $\frac{1}{k+n}\leq\frac{1}{1+n}$ . Therefore, $0<a_{n}\leq \sum_{k=1}^{n}\frac{1}{n+1}==\underset { n }{ \underbrace { \frac { 1 }{ n+1 } +...+\frac { 1 }{ n+1 } } }= \frac{n}{n+1} <1$ .

Thus, it is proved that the increasing sequence is bounded from above, hence the sequence converges.