The Circumference of an Ellipse

Circumference of an Ellipse (Gauss-Kummer)

The Gauss-Kummer Series (Wolfram MathWorld) has better convergence properties than Euler's series:

Circumference S = π*(a+b)(1 + h/2²+ h²/2⁶ + h³/2⁸ + 25h⁴/2¹⁴ + 49h⁵/2¹⁶ + ...)

The coefficient of hⁿ is the square of the fractional binomial coefficient C:

coefficients C_i² terms C_i² hⁱ

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

ε=0.95:
n=10 S=4.41088659279142
n=20 S=4.410886593016652
n=30 S=4.410886593016652
n=50 S=4.410886593016652
n=85 S=4.410886593016652

There are some more series having better convergence properties.