I'm trying to prove the asymptotic statement that for $t\geq 1$:
$\frac{(1+\frac{1}{t})^t}{e} = 1 -\frac{1}{2t} + O(\frac{1}{t^2})$
I know that $(1+\frac{1}{t})^t$ converges to $e$ and the right side looks like the first bit a of the series expansion for $e^{\frac{-1}{2t}}$ but I can't seem to work this thing out. Can someone point me in the right direction here, I'm new to asymptotic analysis and I don't really know how to deftly manipulate big-O notation yet. Thanks.