I have to calculate the limit of the sequence
$a_n := \sum_{k=1}^{n} \frac{1}{n+k}$ .
To do so, I have to show that the following inequation is true:
$\int_{n+1}^{2n+1} \frac{dx}{x} \leqslant a_n \leqslant \int_{n}^{2n} \frac{dx}{x}$
The excercise gives the following hint: "Find appropriate limits for the integrals".
I know, how I calculate the limit of the sequence (without the inequation). But I don't know, how I have to show the inequation. Has anyone a hint witch limits I have to choose to show the inequation? Thanks!
Hint: We have $$\int_{n+1}^{2n+1}\frac{dx}{x}=\sum_{k=1}^{n}\int_{n+k}^{n+k+1}\frac{dx}{x} $$ and $$\int_{n}^{2n}\frac{dx}{x}=\sum_{k=0}^{n-1}\int_{n+k}^{n+k+1}\frac{dx}{x}$$ and $1/x$ is a decreasing function when $x$ grows.