There are several tests for the convergence or divergence of infinite series with all positive terms. We consider two.
If is a function with
a decreasing continuous function
defined for all numbers
, then the infinite series
converges if and only if the integral
converges.
Use the integral test to determine whether or not
converges.
Indeed, it does not as
Does the series
converge?
Consider
. We compute
which is
negative for all
. Thus,
is decreasing.
We compute using integration by parts with so that
and
so that
,
Hence,
converges.
If
and
are two sequences
of positive numbers for which
for every
, then
while
moreover,
.
Does the series
converge?
Yes:
for every
. We
know
. Hence,
converges and is at most
.