Power series in the complementary modulus for the first and second complete elliptic integrals are deduced in terms of binomial series, by exploiting a suitable decomposition of the integration domain. This approach appears to be straightforward, with respect to the standard one. However, despite the procedure is simple, it needs some non-trivial results about binomial series proved in the appendix. Numerical performances of the expansions are also discussed and compared with existing alternative expansions.