Cauchy condensation test

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In mathematics, the Cauchy condensation test is a standard convergence test for infinite series. For a positive monotone decreasing sequence f(n), the sum

\sum_{n=1}^{\infty}f(n)

converges if and only if the sum

\sum_{n=0}^{\infty} 2^{n}f(2^{n})

converges. Moreover, in that case we have

\sum_{n=1}^{\infty}f(n) < \sum_{n=0}^{\infty} 2^{n}f(2^{n}) < 2 \sum_{n=1}^{\infty}f(n).

A geometric view is that we are approximating the sum with trapezoids at every 2n. Another explanation is that, as with the analogy between finite sums and integrals, the 'condensation' of terms is analogous to a substitution of an exponential function. This becomes clearer in examples such as

f(n) = n − a(logn) − b(loglogn) − c.

Here the series definitely converges for a > 1, and diverges for a < 1. When a = 1, the condensation transformation essentially gives the series

\sum n^{-b} (\log n)^{-c}

The logarithms 'shift to the left'. So when a = 1, we have convergence for b > 1, divergence for b < 1. When b = 1 the value of c enters.

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