Qustion
Suppose the following holds: $$ \mathbb E[e^X] = e^\theta \sum_{k=0}^\infty \frac{1}{n^k}\cdot\frac{1}{2^k k!}. $$
Then, We want to show $$ \mathbb E[e^X - e^\theta] = \frac{e^\theta}{2n} + O\left(\frac{1}{n^2} \right). $$
Please give me a road map to show this.
What I know
・The RHS of the assumed equation is a power series in $1/n$.
・According to the reference, it seems to be shown using Taylor's theorem.
・I know Taylor's theorem, but I don't know where to apply it in this case.