I was trying to find a closed form for the sum $\sum_{k=0}^\infty \binom{2n+k+1}{n}/2^{2n+k+1}$.
According to Wolfram
https://www.wolframalpha.com/input/?i=sum+(2n%2Bk%2B1)!%2F(n!*n%2Bk%2B1)!*2%5E(2n%2Bk%2B1))+from+k%3D0+to+infinity
this sum evaluates to 1, but I can't figure out how to prove this.
Any hints.
Recall that $$ \binom nm=\frac1{2\pi i}\oint_{|z|=\rho}\frac{(1+z)^n}{z^{m+1}}dz.\tag1 $$ Therefore, assuming $\rho<1$: $$\begin{align} \sum_{k=0}^\infty \binom{2n+k+1}{n}\left(\frac12\right)^{2n+k+1} &=\sum_{k=0}^\infty\left(\frac12\right)^{2n+k+1}\frac1{2\pi i}\oint_{|z|=\rho}\frac{(1+z)^{2n+k+1}}{z^{n+1}}dz\tag2\\ &=\frac1{2\pi i}\oint_{|z|=\rho}\left(\frac{1+z}2\right)^{2n+1}\frac{dz}{z^{n+1}} \sum_{k=0}^\infty\left(\frac{1+z}2\right)^{k}\tag3\\ &=\frac1{2\pi i}\oint_{|z|=\rho}\left(\frac{1+z}2\right)^{2n+1} \frac1{1-\frac{1+z}2}\frac{dz}{z^{n+1}}\tag4\\ &=\frac1{2\pi i}\oint_{|z|=\rho}\left(\frac{1+z}2\right)^{2n+1} \frac2{1-z}\frac{dz}{z^{n+1}}\tag5\\ &=\operatorname{Res}_{z=0}\left(\frac12\right)^{2n}\frac{1}{z^{n+1}} \sum_{l=0}^{2n+1}\binom{2n+1}l z^l\sum_{k=0}^\infty z^k\tag6\\ &=\left(\frac12\right)^{2n}\sum_{k=0}^n\binom{2n+1}{n-k}\tag7\\ &=\left(\frac12\right)^{2n}2^{2n}=1.\tag8 \end{align} $$
Explanations:
$(1)$ Follows from the residue theorem, since $\binom nm$ is the coefficient at $z^{-1} $ in the Laurent expansion of the integrand about $z=0$.
$(2)$ The binomial coefficient is replaced according to $(1)$.
$(3) $ The terms are rearranged and the order of integration and summation is interchanged (which is possible due to $\rho <1$).
$(4) $ The geometric series is evaluated (which converges due to $\rho <1$).
$(5) $ The result for the geometric series is rearranged.
$(6) $ The residue theorem is applied. The terms $(1+z)^{2n+1}$ and $\dfrac1 {1-z}$ are expanded to binomial sum and geometric series, respectively.
$(7) $ The residue, i.e. the coefficient at $z^{-1} $, is evaluated.
$(8) $ The sum of the binomial coefficients is evaluated to $2^{2n}$ (since $2n+1$ is odd and exactly half of binomial coefficients is summed) which gives rise to the final result.
Preliminary
$$
\begin{align}
a_n
&=\sum_{k=0}^n\frac{\binom{k+n}{k}}{2^k}\tag{1a}\\
&=\sum_{k=0}^n\frac{\binom{k+n-1}{k-1}+\binom{k+n-1}{k}}{2^k}\tag{1b}\\
&=\sum_{k=0}^{n-1}\frac{\binom{k+n}{k}}{2^{k+1}}+\sum_{k=0}^n\frac{\binom{k+n-1}{k}}{2^k}\tag{1c}\\
&=\frac12a_n-\frac{\binom{2n}{n}}{2^{n+1}}+a_{n-1}+\frac{\binom{2n-1}{n}}{2^n}\tag{1d}\\[3pt]
&=\frac12a_n+a_{n-1}\tag{1e}\\[9pt]
&=2a_{n-1}\tag{1f}
\end{align}
$$
Explanation:
$\text{(1a)}$: define $a_n$
$\text{(1b)}$: Pascal Identity
$\text{(1c)}$: substitute $k\mapsto k+1$ in the left sum
$\text{(1d)}$: apply $\text{(1a)}$
$\text{(1e)}$: cancel terms
$\text{(1f)}$: $2$ times $\text{(1e)}$ minus $\text{(1a)}$
Since $a_0=1$, we get $$ \sum_{k=0}^n\frac{\binom{k+n}{k}}{2^k}=2^n\tag2 $$
Answer
$$
\begin{align}
\sum_{k=0}^\infty\frac{\binom{2n+k+1}{n}}{2^{2n+k+1}}
&=\sum_{k=0}^\infty\frac{\binom{2n+k+1}{n+k+1}}{2^{2n+k+1}}\tag{3a}\\
&=\frac1{2^n}\sum_{k=0}^\infty(-1)^{n+k+1}\frac{\binom{-n-1}{n+k+1}}{2^{n+k+1}}\tag{3b}\\
&=\frac1{2^n}\sum_{k=n+1}^\infty(-1)^k\frac{\binom{-n-1}{k}}{2^k}\tag{3c}\\
&=\frac1{2^n}2^{n+1}-\frac1{2^n}\sum_{k=0}^n(-1)^k\frac{\binom{-n-1}{k}}{2^k}\tag{3d}\\
&=2-\frac1{2^n}\sum_{k=0}^n\frac{\binom{k+n}{k}}{2^k}\tag{3e}\\[9pt]
&=1\tag{3f}
\end{align}
$$
Explanation:
$\text{(3a)}$: symmetry of Pascal's Triangle
$\text{(3b)}$: negative binomial coefficient
$\text{(3c)}$: substitute $k\mapsto k-n-1$
$\text{(3d)}$: Binomial Theorem
$\text{(3e)}$: negative binomial coefficient
$\text{(3f)}$: apply $(2)$
