Cauchy Condensation Test

Suppose

is a decreasing sequence of positive terms. Then the series

converges if and only if the series

converges.

Examples:

Use the Cauchy Condensation criteria to answer the following questions:
1. In the sum , list the terms a₄, a_k, and a_2^k. Then show that this series (called the harmonic series) diverges.
2. For which p does the series converge or diverge ? (In addition to the p-Series test , recall the Geometric Series Test for this example)

Proof:

Assume that converges: We have

2^k-1 a_2^k = a_2^k + a_2^k + a_2^k + ... + a_2^k

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.

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