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1 + 1 + 1 + 1 + · · ·

1 + 1 + 1 + 1 + · · ·, also written , is a divergent series.

Where it occurs in physical applications, 1 + 1 + 1 + 1 + · · · may sometimes be interpreted by zeta function regularization. It is the value at s = 0 of the Riemann zeta function

in which the function is defined where the series diverges by analytic continuation. In this sense 1 + 1 + 1 + 1 + · · · = ζ(0) = −¹⁄₂.

Emilio Elizalde presents an anecdote on attitudes toward the series:

In a short period of less than a year, two distinguished physicists, A. Slavnov and F. Yndurain, gave seminars in Barcelona, about different subjects. It was remarkable that, in both presentations, at some point the speaker addressed the audience with these words: 'As everybody knows, 1 + 1 + 1 + · · · = −¹⁄₂'. Implying maybe: If you do not know this it is no use to continue listening.^[1]

See also

• Grandi's series

References