Verifying the formula for the perimeter of an ellipse

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I've been working on the formula for the perimeter of an ellipse, but wasn't able to validate the expression I got while researching online. I'm looking for someone on this forum to perhaps help me with that

The expression I got for the perimeter of an ellipse with semi-major axis a and semi-minor axis b is

$$4\frac{aπ}{2}\sum_{n=0}^\infty \left(\frac{(2n)!}{2^{2n}(n!)^2}\right)^2\frac{\left(\frac{\sqrt{a^2-b^2}}{a}\right)^{2n}}{1-2n}$$

Is this accurate?