How do you use the integral test to determine if #ln2/2+ln3/3+ln4/4+ln5/5+ln6/6+...# is convergent or divergent?
1 Answer
It is divergent. See explanation.
Explanation:
First write the series:
#ln2/2+ln3/3+ln4/4+...=sum_(n=2)^ooln(n)/n#
Before getting into the integral test, we must assure two things first: for the integral test to apply to
Both of these are true since
Furthermore,
So, we see the integral test applies. The integral test states that if the two aforementioned conditions are met, then for
If the integral converges to a real, finite value, then the series converges. If the integral diverges, then the series does too.
So, we take the integral
#int_2^ooln(x)/xdx=lim_(brarroo)int_2^bln(x)/xdx#
Letting
#=lim_(brarroo)int_ln(2)^ln(b)ucolor(white).du#
#=lim_(brarroo)[1/2u^2]_ln(2)^ln(b)#
#=lim_(brarroo)1/2ln^2(b)-1/2ln^2(2)#
As
#=oo#
The integral diverges. Thus, we see that