Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$ I'm undergraduate student of mathematics. 
I need to prove:
$$\sum_{k=0}^{n} \binom{2n}{2k}= 2^{2n-1}$$
Can you please help me

$$\sum_{k=0}^{n} \binom{2n}{2k} = \sum_{k=0}^n \left[\binom{2n - 1}{2k} + \binom{2n - 1}{2k - 1}\right] = \sum_{k=-1}^{2n} \binom{2n - 1}{k} = \sum_{k=0}^{2n - 1} \binom{2n - 1}{k} = 2^{2n-1} $$
