Question

Question about the divergence test for proving divergence.

Why can't i prove divergence of the Series (n+1)/n(n+2) using the divergence test. You have to use the integral test. If i just use l'hospitals rule on the series, I get that the series converges.

yet if you use the divergence test on a series such as n^2/2n^2 you can easily prove divergence.

Answer 1

\[\sum_{1}^{\inf}n+1/n(n+2)\]

Answer 2

\[\sum_{n=1}^\infty\frac{n+1}{n(n+2)}\]
\[\lim_{n\to\infty}\frac{n+1}{n(n+2)}=\lim_{n\to\infty}\frac{1+\frac{1}{n}}{n+2}=0\]
The divergence test says that if the limit is non-zero, then the series diverges. Here, the limit is 0. This doesn't tell you the series converges, though.

I think you're confusing the following statement: If \(\sum a_n\) is convergent, then \(\lim_{n\to\infty}a_n=0\). However, the converse is not true: If \(\lim_{n\to\infty}a_n=0\), then \(\sum a_n\) does not necessarily converge.

Answer 3

In short, the n-th term test for divergence is inconclusive, and so you have to use a different test.

Answer 4

ah, your right.