TITLE: determining if sequence has upper bound
QUESTION [5 upvotes]: I am somewhat stuck in my calculations when determining if sequence has an upper bound.
The sequence $$x_n = \frac{1}{n+1}+\frac{1}{n+2}+..+\frac{1}{2n-1}+\frac{1}{2n}$$ 
Is equal to $$\frac{1}{n}(\frac{1}{1+\frac{1}{n}}+\frac{1}{1+\frac{2}{n}}+..+\frac{1}{1+\frac{n}{n}})$$
And so I notice that all the denominators are greater than 1, which means that all terms in the parentheses are less than 1. 
But how can I determine further if there is an upper bound?

REPLY [4 votes]: Notice the Riemann sum
$$\frac1n\sum_{k=1}^n \frac1{1+k/n} < \int_0^1\frac{dt}{1+t} = \log 2$$