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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 p-series converges if p > 1.

Example 4.2.8:
 
  • Use the Cauchy Condensation criteria to answer the following questions:
    1. In the sum , list the terms a4, ak, and a2k. 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

2k-1 a2k = a2k + a2k + a2k + ... + a2k
     
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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