Theorem 18

If m is a positive finite integer,

$$1 - I_w (m) = e^{-w} \sum^{m-1}_{k=0} \frac{w^k}{k!}.$$

Proof: By Definition 9,

$$I_w (m) = \frac1{(m-1)!} \int^w_0 e^{-s} m^{m-1} ds.$$

Integrating by parts so that the power of s is decreased by one at each step,

$$\begin{array} {r@{~}c@{~}l} I_w (m) & = & \displaystyle \fra... ...-2}}{(m-2)!} + \ldots + \frac{w^0}{0!} \right\} + 1.\end{array}$$

Hence,

$$1 - I_w (m) = e^{-w} \sum^{m-1}_{k=0} \frac{w^k}{k!}.$$