コーシーの凝集判定法
∑ n = 0 ∞ 2 n f ( 2 n ) = f ( 1 ) + ( f ( 2 ) + f ( 2 ) ) + ( f ( 4 ) + f ( 4 ) + f ( 4 ) + f ( 4 ) ) + ⋯ = ( f ( 1 ) + f ( 2 ) ) + ( f ( 2 ) + f ( 4 ) + f ( 4 ) + f ( 4 ) ) + ⋯ ≤ ( f ( 1 ) + f ( 1 ) ) + ( f ( 2 ) + f ( 2 ) + f ( 3 ) + f ( 3 ) ) + ⋯ = 2 ∑ n = 1 ∞ f ( n ) {\displaystyle {\begin{array}{rcl}\sum \limits _{n=0}^{\infty }2^{n}f(2^{n})&=&f(1)+{\Big (}f(2)+f(2){\Big )}+{\Big (}f(4)+f(4)+f(4)+f(4){\Big )}+\cdots \\&=&{\Big (}f(1)+f(2){\Big )}+{\Big (}f(2)+f(4)+f(4)+f(4){\Big )}+\cdots \\&\leq &{\Big (}f(1)+f(1){\Big )}+{\Big (}f(2)+f(2)+f(3)+f(3){\Big )}+\cdots \\&=&2\sum \limits _{n=1}^{\infty }f(n)\end{array}}}