text: Divergence test for series | Physics Forums
  Homework Help: Divergence test for series
  Not really a problem, more of a general question. When exactly can you use the Divergence test. Does it only work on both series and sequences?
  The series Diverges if lim ƩAn ≠ 0
  If you take the lim of the series n^3/2n^3 ≠ 0 there it diverges.
  Now, look at the series (n+1)/n(n+2). You have to use the integral test to show convergence or divergence for this. After doing it, you get the series Diverges. Why can't i just use l'hospitals rule on the 2nd series, and get 1/2n, then take the limit. And it should converge to zero.
  Is it because i used l'hospitals rule therefore i can not use the divergence test anymore?
  steel1, Apr 23, 2013
  NO, the series diverges if ##\lim_{n\to\infty}A_n\ne 0## (no sum).
  If the nth term doesn't go to zero the series diverges. But the nth term may go to zero and yet the series diverges anyway. So the test for divergence is sufficient but not necessary for divergence. That's why you need other more sophisticated tests.