Is this positive term series convergent? My series is: 

$\frac{1}{1+2^{-1}}\ +\frac{1}{1+2^{-2}}\ +\ldots$

I see my $nth$ term is $\frac{1}{1+2^{-n}}$

How do I test for its convergence?
A necessary condition for the convergence of $\sum_n a_n$ is that $\lim_n a_n=0$. In your case, $\lim_n a_n=1$ so the series doesn't converge. In fact you can show that it diverges to $+\infty$ (every term in your sequence is greater than $\frac{1}{2}$).
Yes the series convergent to 1.
Series 1/2 power n divergent but 1/2 power n converges to infinity and 1/1+2 power n converges to 1.