Math Help - Perimeter of an Ellipse

1. Perimeter of an Ellipse

I need to know how one would evaluate this expression given:

$h = \frac{(a - b)^2}{(a + b)^2}$

Here's the expression for the major radius a and minor radius b of an ellipse:

$P = \pi (a + b) \sum_{n = 0}^\infty \left[{\frac{1}{2} \choose n}^2 h^n\right]$

Where:

$a = 405407$

$b = 338246$

2. Originally Posted by Aryth
I need to know how one would evaluate this expression given:

$h = \frac{(a - b)^2}{(a + b)^2}$

Here's the expression for the minor radius a and major radius b of an ellipse:

$P = \pi (a + b) \sum_{n = 0}^\infty {\frac{1}{2} \choose n}^2 h^n$

Where:

$a = 405407$

$b = 338246$
Here is what Wikipedia has to say about it:

3. Those are all approximations (With the exception of the sum/product), I'm familiar with those and I have my own approximation whose error cannot exceed:

$\delta = \frac{0.4e^8}{(1 - e^2)}$

Where e is eccentricity and for my particular case, it is:

$e = 0.05$

So, the approximation is very close, yet I wish to have an exact measurement, and according to my research, this is one of four exact methods.

I was wondering how to use the series above. I'm not very experienced with infinite series, which is why I posted it.

4. Originally Posted by Aryth
Those are all approximations (With the exception of the sum/product), I'm familiar with those and I have my own approximation whose error cannot exceed:

$\delta = \frac{0.4e^8}{(1 - e^2)}$

Where e is eccentricity and for my particular case, it is:

$e = 0.05$

So, the approximation is very close, yet I wish to have an exact measurement, and according to my research, this is one of four exact methods.

I was wondering how to use the series above. I'm not very experienced with infinite series, which is why I posted it.
It is exact untill you want to evaluate it when (except for a few special cases) you will have to used an approximation to the sum. There is no general finite closed form expression for this in terms of elementary functions

RonL