Finding the sum of the infinite series $sum^{infty}_{k=1} frac{k-1}{k!}$

Find the sum of the infinite series $$\sum^{\infty}_{k=1} \frac{k-1}{k!}.$$

Using the ratio test, for example, one can easily show that the series is convergent.

But how does one go about finding the sum?

Perhaps the power series will be useful, such as $$e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}.$$

Answer

Sketch:

$$
\sum_{k=1}^\infty \frac{k-1}{k!}=\sum_{k=1}^\infty\frac{k}{k!}-\sum_{k=1}^\infty\frac{1}{k!}=\sum_{k=1}^\infty\frac{1}{(k-1)!}-\sum_{k=1}^\infty\frac{1}{k!}=\sum_{k=1}^\infty\left(\frac{1}{(k-1)!}-\frac{1}{k!}\right)
$$

This is telescoping. Make sure to understand and be able to explain what happens in each equality. Can you take it from here?