Rearranging a summation of factorials into a Taylor series expansion

Question

I've come across this problem which I has truly stumped me. Basically, I'm meant to rearrange the below summation to represent in some way some kind of Taylor series expansion to show that it approximates e.

$$\sum_{n=1}^{26} \frac1{(26-n)!}$$

I have no idea where to start as the only information I can find regarding the rearrangement of factorials/summations refers to either sums to infinity, or expressions with an n value in the numerator as well.

Answer 1

Let $k=26-n$. Now, $k$ goes from $0$ to $25$, and you have $\sum_{k=0}^{25}\frac1{k!}$, which is the partial sum of the first twenty five terms of the Taylor Series for $e^x$, with $x=1$. Note that it doesn't equal $e$, but it is a decent approximation. For it to truly equal $e$, you would have to take the infinite sum, or $\sum_{k=0}^\infty\frac1{k!}$.