I'm having trouble with the following problem:
Use the integral test to find upper and lower bounds on the limit of the series.
$\displaystyle \sum_{k=1}^{\infty} \frac {1}{k^3}$
The integral can be evaluated easily.
$\displaystyle \int \frac {1}{x^3} \, dx = -\frac{1}{2x^2} + C$
Thus
$\displaystyle \int_1^{\infty} \frac{1}{x^3} \, dx < \sum_{k=1}^{\infty} \frac{1}{k^3} < 1 + \int_1^{\infty} \frac{1}{x^3} \, dx$
$\displaystyle \frac12 < \sum_{k=1}^{\infty} \frac{1}{k^3} < \frac32$