During a derivation of handed out lecture notes they have the following $$\sum_{m \geq 0}\left(\begin{array}{l} {n} \\ {m} \end{array}\right)(2 m+1) = 2 \sum_{m \geq 0}\left(\begin{array}{c} {n} \\ {m} \end{array}\right) m+2^{n}$$
But i cant quite see why this is correct. I see how you might rewrite it as $$2 \sum_{m \geq 0}\left(\begin{array}{c} {n} \\ {m} \end{array}\right) m+\left(\begin{array}{c} {n} \\ {m} \end{array}\right)$$ But then the binomial coefficient would have to be $2^n$. Which only makes sense if they mean that $$\sum_{m \geq 0}\left(\begin{array}{l} {n} \\ {m} \end{array}\right)(2 m+1) = 2 \sum_{m \geq 0}\left(\left(\begin{array}{c} {n} \\ {m} \end{array}\right) m\right)+2^{n}$$ What am i missing here?
$\sum\limits_{m = 0}^n \binom{n}{m}$ is indeed equal to $2^n$. This can be seen by considering $2^n = (1+1)^n$ and then using the binomial theorem.