Cauchy condensation test

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

:

A geometric view is that we are approximating the sum with trapezoids at every $2^\{n\}$. 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

:$f(n)\; =\; n^\{-a\}\; (log\; n)^\{-b\}\; (log\; log\; n)^\{-c\}$.

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.

External links

* [http://pirate.shu.edu/projects/reals/numser/t_conden.html Cauchy condensation test proof]