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Divergent geometric series : ウィキペディア英語版
Divergent geometric series
In mathematics, an infinite geometric series of the form
:\sum_^\infty ar^k = a + ar + ar^2 + ar^3 +\cdots
is divergent if and only if | ''r'' | ≥ 1. Methods for summation of divergent series are sometimes useful, and usually evaluate divergent geometric series to a sum that agrees with the formula for the convergent case
:\sum_^\infty ar^k = \frac.
This is true of any summation method that possesses the properties of regularity, linearity, and stability.
==Examples==
In increasing order of difficulty to sum:
*1 − 1 + 1 − 1 + · · ·, whose common ratio is −1
*1 − 2 + 4 − 8 + · · ·, whose common ratio is −2
*1 + 2 + 4 + 8 + · · ·, whose common ratio is 2
*1 + 1 + 1 + 1 + · · ·, whose common ratio is 1.

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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