Sphere function: $S_n = \frac{2 \pi^{n/2}}{ (\frac{1}{2} n - 1)! }$
Sphere volume: $V_n = \frac{S_n R^n}{n}$
Sphere surface: $A_n = \frac{dV_n}{dR} = S_n R^{n-1}$
$S_n = \frac{2\pi^{n/2}}{\Gamma(n/2)}$ for the $\Gamma$ function definition
$S_n = \frac{2^{(n+1)/2} \pi^{(n-1)/2}}{(n-2)!!}$ for n odd
$S_n = \frac{2\pi^{n/2}}{(n/2-1)!}$ for n even

Sum of surface areas of all even dimensions of a particular radius:
$\Sigma_{n=2,4,...}^\infty A_n$
$= \Sigma_{n=2,4,...}^\infty S_n R^{n-1}$
$= \Sigma_{n=2,4,...}^\infty \frac{2\pi^{n/2}}{(n/2-1)!} R^{n-1}$
$= \Sigma_{k'=1}^\infty \frac{2\pi^{k'}}{(k'-1)!} R^{2k'-1}$ for $k'=n/2$, $n=2k'$
$= \Sigma_{k=0}^\infty \frac{2\pi^{k+1}}{k!} R^{2k+1}$ for $k=k'-1$, $k'=k+1$
$= 2 \pi R \; \Sigma_{k=0}^\infty \frac{(\pi R^2)^k}{k!}$
$= 2 \pi R \; exp(\pi R^2)$ by $exp(x) = \Sigma_{k=0}^\infty \frac{x^k}{k!}$

Sum of surface areas of all odd dimensions of a particular radius:

Integral of surface areas of all dimensions $\ge 2$:
$\int_{n=2}^{n=\infty} A_n dn$
$=\int_{n=2}^{n=\infty} S_n R^{n-1} dn$
$=\int_{n=2}^{n=\infty} \frac{2\pi^{n/2}}{\Gamma(n/2)} R^{n-1} dn$
$=\frac{4\pi R}{\pi R^2} \int_{k=1}^{k=\infty} \frac{(\pi R^2)^k}{\Gamma(k)} dk$ for $k=n/2$, $n=2k$, $dn=2dk$

Ramanujan says
$\int_{x=0}^{x=\infty} \frac{w^x}{\Gamma(x+1)} dx = e^w - \int_{y=-\infty}^{y=\infty} \frac{exp(-we^y)}{y^2+\pi^2} dy$
$= \frac{1}{w} \int_{y=1}^{y=\infty} \frac{w^y}{\Gamma(y)} dx$ for $y=x+1$, $x=y-1$, $dx=dy$

$=4 \pi R \left( exp(\pi R^2) - \int_{y=-\infty}^{y=\infty} \frac{exp(-\pi R^2 e^y)}{y^2+\pi^2} dy \right)$