Integral Test: Let

from k=1 to infinity be a series with positive terms and define a function f on [1,

) so that f(k)=

for each positive integer k. Suppose that f is continuous and decreasing on [1,

) and that the limit of f(x) as x approaches infinity = 0. Then the series

from k=1 to infinity converges if and only if the improper integral

f from 1 to infinity converges. Furthermore, if S is the sum of the series and

is the nth partial sum of the series, then 0 < S-

<

f from n to infinity for each positive integer n.

Prove the integral test.