Prove:

The sequence An = 1/(n + 1) + 1/(n + 2) +...+ 1/(2n) has a limit.

Printable View

- Sep 10th 2009, 07:20 PMcgiulzLimit of a simple sequence
Prove:

The sequence An = 1/(n + 1) + 1/(n + 2) +...+ 1/(2n) has a limit. - Sep 10th 2009, 08:56 PMJG89
$\displaystyle n + 1 < n + 2 < \cdots < 2n \Leftrightarrow \frac{1}{2n} < \frac{1}{2n - 1} < \cdots < \frac{1}{n+1} $.

Notice that your sum has n amount of terms, and so $\displaystyle 0 < \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} < \frac{n}{2n} = \frac{1}{2} $.

Now show that it's monotonic, and you're finished. - Sep 10th 2009, 09:11 PMcgiulz
Thanks allot for your help. (Happy)

Quick question:

Would it not be <=? - Sep 11th 2009, 05:55 AMKrizalid
you're working for $\displaystyle n\ge1,$ so everything is positive.

- Sep 11th 2009, 06:04 AMcgiulz
Really I was referring to the <= n/2n = 1/2. This still holds though?

Thanks