Comparison Test

Suppose that

converges absolutely, and

is a sequence of numbers for which

| b_n | | a_n | for all n > N

Then the series

converges absolutely as well.

If the series converges to positive infinity, and is a sequence of numbers for which

for all n > N

Then the series

also diverges.

This is a useful test, but the limit comparison test, which is rather similar, is a much easier to use, and therefore more useful. However, this comparison test is very easy to memorize: Assuming that everything is positive, for simplicity, say we know that:

| b _n | | a _n | for all n

then just sum both sides to see what you get formally:

Then:

If the left sides equals infinity, so must the right side.
If the right side is finite, the left side also converges.

Examples:

Does converge or diverge ?
Does converge or diverge ?

Proof:

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 S_N

Since 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: