How to get a partial sum formula

Question

Let S denote sum from 1 to n of (k-1)/k! . I tried obtaining a partial sum formula, but I couldn't get too far. WolframAlpha comes with quite a simple form, but I fail to see how they got there . Can anyone suggest a solution ?

Answer 1

For integer $k\ge1,$ $$\frac{k-1}{k!}=\frac k{k\cdot (k-1)!}-\frac1{k!}=\frac1{(k-1)!}-\frac1{k!}$$ which is the $k$th term of a Telescoping series

Answer 2

$$\frac{k-1}{k!} = \frac{1}{(k-1)!} - \frac{1}{k!}$$ So your sum is (this is a telescoping sum) $$\sum_{k=1}^{n} \left(\frac{1}{(k-1)!} - \frac{1}{k!} \right) = \left(\frac{1}{0!} - \frac{1}{1!}\right) + \left(\frac{1}{1!} - \frac{1}{2!}\right) + ... \left(\frac{1}{(n-1)!} - \frac{1}{n!}\right) = 1 - \frac{1}{n!}$$