How do you use the Nth term test on the infinite series #sum_(n=2)^oon/ln(n)# ?

Question

How do you use the Nth term test on the infinite series  #sum_(n=2)^oon/ln(n)# ?

Luke Alvarez · Accepted Answer

The Nth Term Test is a basic test that can help us figure out if an infinite series is divergent. It states that if the #lim_(n->oo)# of our series is not equal to #0#, the series is divergent. Note that this does not mean that if the #lim_(n->oo) = 0#, the series is convergent, only that it might converge. All we can tell from this test is whether or not it diverges.

Using this test with our series, we have:

#lim_(n->oo) n/ln(n)#

If we replace #n# with #oo#, we end up with:

#oo/ln(oo) = oo/oo#

Since we have an Indeterminate Form of #oo/oo#, we can apply L'Hôpital's rule, which says that if we end up with #oo/oo# or #0/0#, we can then take the #lim# of the derivative of the numerator over the derivative of the denominator. Since the derivative of #n# is #1#, and the derivative of #ln(n)# is #1/n#, we have:

#lim_(n->oo) 1/(1/n) = lim_(n->oo) n = oo#

Because we end up with a non-zero answer, we know that the series diverges.