Question on the Cauchy Condensation test

by Simfish
Tags: cauchy, condensation, test
Notice that the function f in $\sum_{n=0}^{\infty} 2^{n}f(2^{n})$ need not be well-defined for all arguments in the real numbers, but only for the natural numbers (including the zero).
What you have is a positive monotone decreasing sequence (which is denoted here by $f(n)$ but could just as well be written as $a_n$). Now consider the sum (infinite series): the idea now is to form summation blocks with length $2^n$ and find a useful estimate for each block. Every block incorporates $2^n$ summands and the biggest summand in each block is the first one, as the sequence is monotone decreasing.