Using the integral test, how do you show whether #sum 1/(n(lnn)^2) # diverges or converges from n=1 to infinity?
1 Answer
This may be a "trick question". The term
Explanation:
I don't know whether this is a trick question the student was asked of if there is an error in posting it here.
If we eliminate the first term and do the integral test for
I think it is fairly clear that the function
# = lim_(brarroo)int_2^b 1/(lnx)^2 1/x dx#
# = lim_(brarroo)int_2^b (lnx)^-2 1/x dx#
# = lim_(brarroo) [-(lnx)^-1]_2^b#
# = lim_(brarroo) [-1/lnb - (-1/ln2)]#
# = 0+1/ln2#
So the integral and the series both converge.
Reminder
If we really wanted to we could integrate:
for
The integral