Thread: nth term test question.

This problem diverges: lim n->infinity (k^2) / (k^2 -1)

this diverges by the nth term test. =/ 0 the limit equals 1.

however, my question is, since this equals 1. Why does this not converge to 1?

is this because it doesn't converge to 1 quick enough?

Originally Posted by rcmango

This problem diverges: lim n->infinity (k^2) / (k^2 -1)

this diverges by the nth term test. =/ 0 the limit equals 1.

however, my question is, since this equals 1. Why does this not converge to 1?

is this because it doesn't converge to 1 quick enough?

It's simple to see that the SEQUENCE converges to 1. eg. .

The nth term test is used to test whether a SERIES, not a sequence, diverges!! And clearly the series diverges (the sum of 1's --> oo). Unsurprising since the nth term does not approach zero.

Or are you asking for a proof of the nth term (limit) test? If so, consult any standard reference on the topic .....

Also we have .
Adding infinitely many ones diverges.

Okay, didn't know it was testing a series, i see now. thanks.

it doesn't converge to one because in a geometric series, if a sub n doesn't equal 0, then it doesn't converge at all. Therefore, the sum might be 1, but that doesn't mean it converges to 1.