Generating Series for the set of all compositions which have an even number of parts. 


I'm having trouble showing that the generating series for all compositions which have an even number of parts.
I'm given that each part congruent to 1 mod 5 is equal to:$$\frac{1-2x^5+x^{10}}{1-x^2-2x^5+x^{10}}$$
If you could help me out that would be great!

Here's another method.  First, you should convince yourself that the generating function for compositions with $k$ parts is given by
$$(x + x^2 + x^3 + \ldots)^k.$$  This is because choosing a composition $k_1 + k_2 + \ldots + k_m$ corresponds to choosing $x^{k_1}$ in the first factor, $x^{k_2}$ in the second, and so on.  This is the same way you would multiply out the power series - choosing every possible $k$-tuple of terms, one from each factor, and multiplying them, and then summing the result.  
Now simplify this expression and sum this over all even $k$ - it will become a nice rational generating function.
