The Circumference of an Ellipse

Circumference of an Ellipse (Euler)

In a paper, published in 1738, Euler found a series for the arc length of an ellipse:

Circumference S = 2π*a*(1 + c₁ ε² + c₂ ε⁴ + c₃ ε⁶ + c₄ ε⁸ + ...)

coefficients c_i terms c_i ε²ⁱ

The convergence of the series is very slow when the eccentricity ε is almost 1:

ε=0.95:
n=10 S=4.423712404139952
n=20 S=4.412502156808594
n=30 S=4.411186402526338
n=50 S=4.410902681269847
n=85 S=4.410886764405142

There are some more series having better convergence properties.