From a fraction with infinite sum in denominator to partial fractions?

[megzaz]

From a fraction with infinite sum in denominator to partial fractions??

I am currently studying a course on Perturbation Methods and in particular an example considering the following integral $$\int_{0}^{\frac{\pi}{4}} \frac{d\theta}{\epsilon^2 + \sin^2 \theta}.$$

There's a section of the working where, having used the Taylor expansion of sin near 0 and using sin θ ≈ θ together with substitution θ=εu, we get the following fraction for the integrand
$$\frac{1}{\epsilon^2 + \epsilon^2 u^2 - \frac13 \epsilon^4 u^4 + \cdots}.$$
This then in both my lecture notes and a book I'm following becomes
$$\frac{1}{\epsilon^2}\left( \frac{1}{1+u^2} + \frac{\epsilon u^4}{3(1+u^2)^2} + \cdots \right).$$

Can anyone see how these are equal?

 

[mathman]

It looks like using a variation on 1/(1-x) = 1 + x + x² + ... after factoring out 1/{ε²(1+u²)}.

Also second term numerator should be (εu)⁴