Cauchy Condensation Test
Suppose
is a decreasing sequence of positive terms. Then the series
converges if and only if the series
converges.
This test is rather specialized, just as Abel's Convergence Test. The main purpose of the Cauchy Condensation test is to prove that the pseries converges if p > 1.
Example 4.2.8:  

Proof:
Assume that converges: We have
2^{k1} a_{2k} = a_{2k} + a_{2k} + a_{2k} + ... + a_{2k}because the sequence is decreasing. Hence, we have that
Now the partial sums on the right are bounded, by assumption. Hence the partial sums on the left are also bounded. Since all terms are positive, the partial sums now form an increasing sequence that is bounded above, hence it must converge. Multiplying the left sequence by 2 will not change convergence, and hence the series converges.
Assume that converges: We have
Therefore, similar to above, we get:
But now the sequence of partial sums on the right is bounded, by assumption. Therefore, the left side forms an increasing sequence that is bounded above, and therefore must converge.