Newton's pi series is the rapidly convergent power series for pi that Newton obtained in 1666 by expanding the area
of a circular segment. For a circle of unit diameter,
the area under
from 0 to
combines with the adjacent triangle area to give
Expanding the integrand using the binomial series
and integrating term by term gives
The choice of the small circular segment makes the powers decrease quickly, so the method was much faster than the polygonal approximations used previously (Borwein
et al. 1989; Borwein and Bailey 2003, pp. 105-106; Veritasium 2021).
from 0 to 
combines with the adjacent triangle area to give
