I only care about integers $\geq 1$.
It looks trivial (and probably is), but I can't seem to find an answer except by plotting the graph. I've tried to show $f(1)=\frac{1}{2}$, then taking the derivative and showing it is positive, but the derivative turns out to be quite complicated. I've tried to approximate $f$ as $e^x$ somehow, but the inequality $1+x\leq e^x$ is in the wrong direction to be useful. I've tried to apply the general binomial theorem and separating the even and odd terms, but didn't know what do to with it. What else could I try?