# Limit of a simple sequence

• Sep 10th 2009, 07:20 PM
cgiulz
Limit of a simple sequence
Prove:

The sequence An = 1/(n + 1) + 1/(n + 2) +...+ 1/(2n) has a limit.
• Sep 10th 2009, 08:56 PM
JG89
$n + 1 < n + 2 < \cdots < 2n \Leftrightarrow \frac{1}{2n} < \frac{1}{2n - 1} < \cdots < \frac{1}{n+1}$.

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

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

Quick question:

Would it not be <=?
• Sep 11th 2009, 05:55 AM
Krizalid
you're working for $n\ge1,$ so everything is positive.
• Sep 11th 2009, 06:04 AM
cgiulz
Really I was referring to the <= n/2n = 1/2. This still holds though?

Thanks