The following function is defined for all
, and
is a positive integer. Prove or provide a counterexample to the following statement.
If is positive and if
then
Incomplete.
The following function is defined for all
, and
is a positive integer. Prove or provide a counterexample to the following statement.
If is positive and if
then
Incomplete.
After looking at the counterexample for exercise 25, I’ll have to rework this one because
is not a necessary condition for convergence. There’s actually a pretty simple proof for this one since
is set to be positive for all 
Proof. If
is positive for all
, then, for any
for some integer
, the integral
is bounded above by
. Thus, if
as
, then so does
.
Proof. Since
takes positive values for all
, for
to converge as
,
must go to zero as
. But if
as
and
, then, following the proof written in exercise 21,
converges and has value A.
I think I broke the Latex interpreter… lol. That mess of script is supposed to read: