Grandi's series is the divergent series
Its partial sums alternate between 1 and 0, so the series does not converge in the ordinary sense. The
arithmetic means of the partial
sums converge to 
, and the associated power
series satisfies
whose limit as 
is also 
. Thus Grandi's series is both
Cesàro summable and Abel
summable to 
.
See also
Abel-Regularized Sum,
Arithmetic Mean,
Cesàro
Sum,
Convergence,
Divergent
Series
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References
Hardy, G. H. Divergent Series. Providence, RI: Amer. Math. Soc., 1991.Numberphile.
"One Minus One plus One Minus One." Jun. 25, 2013. https://www.youtube.com/watch?v=PCu_BNNI5x4.
Cite this as:
Weisstein, Eric W. "Grandi's Series."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GrandisSeries.html
Subject classifications
