Can you not apply $\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}$ to the series $\sum^{n+1}_{k=0}\binom{n+1}{k}$

Question

If you have the series $\sum^{n+1}_{k=0}\binom{n+1}{k}$, and apply $\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}$, then it would become $\sum^{n+1}_{k=0}\binom{n}{k}+\sum^{n+1}_{k=0}\binom{n}{k-1}$, but $\binom{n}{n+1}$ is not defined, and it is the last term in $\sum^{n+1}_{k=0}\binom{n}{k}$, so what happens then? Is this a valid thing to do?