When Is the Divergence Test Applicable for Series?

[steel1]

Homework Statement

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?

Homework Equations

The series Diverges if lim ƩAn ≠ 0

The Attempt at a Solution

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?

 

[LCKurtz]

Homework Statement

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?

Homework Equations

The series Diverges if lim ƩAn ≠ 0

NO, the series diverges if $\lim_{n\to\infty}A_n\ne 0$ (no sum).

The Attempt at a Solution

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?

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.