1 + 1 + 1 + 1 + ⋯
In mathematics,, also written,, or simply, is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers. The sequence 1 can be thought of as a geometric series with the common ratio 1. Unlike other geometric series with rational ratio, it converges in neither the real numbers nor in the -adic numbers for some . In the context of the extended real number line
since its sequence of partial sums increases monotonically without bound.
Where the sum of occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at of the Riemann zeta function
The two formulas given above are not valid at zero however, so one might try the analytic continuation of the Riemann zeta function,
Using this one gets,
where the power series expansion for about follows because has a simple pole of residue one there. In this sense.
Emilio Elizalde presents a comment from others about the series:
