Dynamic Programming Euler's Number

The mathematical constant $e$ (Euler's number) is defined as follows:

$$e=\sum _{i=0}^{\mathrm{\infty }}\frac{1}{i!}=\frac{1}{1}+\frac{1}{1}+\frac{1}{1\cdot 2}+\frac{1}{1\cdot 2\cdot 3}+\frac{1}{1\cdot 2\cdot 3\cdot 4}+\dots$$

I've provided an implementation of a function to approximate $e$ up to a number of $n$ terms in code.js for you. However, it does repeated work -- identify where the repeated work occurs and avoid it through bottom-up dynamic programming (not memoization). Change the implementation accordingly.

Test your new functions; I've provided some basic testing code in code.test.js but you need to go beyond that.

Runtime Analysis

What is the worst-case big $\mathrm{\Theta }$ time complexity of your algorithm?