How to get the ordinary generating function for this series I came across the following sum:
$$
\sum_{k \geq 0} \frac{2^k}{2^k+1}
$$
Is there a way to derive the ordinary generating function (OGF) for this sum?, i.e. given the series:
$$
A(z) = a_0 + a_1z^1 + a_2z^2 + ... +a^kz^k + ...
$$
if we have for instance $a_k=1$ for all $k \geq 0$, we have OGF $A(z)=1/(1-z)$, or if the $a_k$'s represent the harmonic numbers $H_k$ for $k \geq 0$, we have the OGF $\frac{1}{1-z}\ln{(\frac{z}{1-z})}$.
So is there a way to get the OGF where $a_k=\frac{2^k}{2^k+1}$?

$\lim_{k\to\infty} a_k=1 \not= 0$, so the series diverges.
It sounds like you want to compute the ordinary generating function $$\sum_{k\ge 0} \frac{2^k}{2^k+1}z^k.$$
Applying the ratio test shows that this series converges when 
$$\lim_{k\to\infty}\left|\frac{a_{k+1}}{a_k}\right|=\lim_{k\to\infty}\frac{|(2z)^{k+1}|}{2^{k+1}+1}\cdot\frac{2^k+1}{|(2z)^k|}=|z|<1.$$
