Suppose is a convergent sequence of positive integers. Let
which is the error if you approximate using the first
From Theorem 6.3.2 we get the following.
Suppose is a continuous, positive, decreasing function on
and is convergent. Then
In Theorem 6.3.2
is decreasing and
using the first
terms of the series.
The proposition above with
tells us that
and we hvae
so the integral error bound was really good in this case.
convergers or diverges. Answer: It converges, since