Think about breaking the area under the curve up into rectangles.
Afterall, that is what we are doing when we integrate.
The area of each rectangle can be represented by, say, .
Compare the area under the curve to that of the rectangles.
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.