# Thread: nth term test question.

1. ## 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?

2. 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. $\frac{k^2}{k^2 - 1} = 1 + \frac{1}{k^2 - 1}$.

The nth term test is used to test whether a SERIES, not a sequence, diverges!! And clearly the series $\sum_{n = 2}^{\infty} \left (1 + \frac{1}{k^2 - 1} \right)$ 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 .....

3. Also we have $\sum\limits_{k = 2}^\infty 1 \le \sum\limits_{k = 2}^\infty {\left[ {1 + \frac{1}{{k^2 - 1}}} \right]}$.