Cauchy condensation test について

converges. Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original.
== Estimate ==

The Cauchy condensation test follows from the stronger estimate
:

0 \ \leq\ \sum_^ f(n)\ \leq\ \sum_^ 2^f(2^)\ \leq\ 2\sum_^ f(n)\ \leq\ +\infty

which should be understood as an inequality of extended real numbers. The essential thrust of a proof follows, following the line of Oresme's proof of the divergence of the harmonic series.
To see the first inequality, the terms of the original series are rebracketed into runs whose lengths are powers of two, and then each run is bounded above by replacing each term by the largest term in that run: the first one, since the terms are non-increasing.
:

\begin\sum_^ f(n) & = &f(1) & + & f(2) + f(3) & + & f(4) + f(5) + f(6) + f(7) & + & \cdots \\ & = &f(1) & + & \Big(f(2) + f(3)\Big) & + & \Big(f(4) + f(5) + f(6) + f(7)\Big) & + &\cdots \\ & \leq &f(1) & + & \Big(f(2) + f(2)\Big) & + & \Big(f(4) + f(4) + f(4) + f(4)\Big) & + &\cdots \\ & = &f(1) & + & 2 f(2) & + & 4 f(4)& + &\cdots = \sum_^ 2^ f(2^)\end

To see the second, the two series are again rebracketed into runs of power of two length, but "offset" as shown below, so that the run of

\textstyle2\sum_^f(n)

which ''begins'' with

\textstyle f(2^)

lines up with the end of the run of

\textstyle\sum_^2^f(2^)

which ''ends'' with

\textstyle f(2^)

, so that the former stays always "ahead" of the latter.
:

\begin\sum_^ 2^f(2^) & = & 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_^ f(n)\end

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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