The integral test states that a seriesan converges absolutely if these exists a positive decreasing function f such that f (n) =
an
for each positive integer n and
To verify convergence of
using the
integral test, define f by
f (x) =
and note the
following.
Evaluate, Evaluate Numerically
dx =
= 5.805497209
Thus, this integral is finite. (Although for
f (x) = , it
is true that
f (1) < f (2) < f (3), you can verify that f is decreasing for x > 3. In fact,