4.2. Convergence Tests
If the series converges to positive infinity, and is a sequence of numbers for which an bn for all n > N. Then the series also diverges.
| b n | | a n |for all n. then just sum both sides to see what you get formally:
- If the left sides equals infinity, so must the right side.
- If the right side is finite, the left side also converges.
The proof, at first glance, seems easy: Suppose that converges absolutely, and | b n | | a n | for all n. For simplicity, assume that all terms in both sequences are positive. Let
S N = and T N =Then we have that
T N SNSince the left side is a convergent sequence, it is in particular bounded. Hence, the right side is also a bounded sequence of partial sums. Therefore it converges.
This proof wrong, because it does show that the sequence of partial sums is bounded but it is not necessarily true that a bounded series also converges - as we know.
However, this proof, slightly modified, does work: Again, assume that all terms in both sequences are positive. Since converges, it satisfies the Cauchy criterion:
| | <if m > n > N. Since | b n | | a n | we then have
| | | | <if m > n > N. Hence satisfies the Cauchy criterion, and therefore converges.
The proof for divergence is similar.