How do you take the derivative when there is a summation operator in this step..
$$\frac{d}{dt} \left[1-\sum_{n=0}^{k-1} \frac{(\lambda t)^n e^{-\lambda t}}{n!} \right] = \lambda e^{-\lambda t} \left(\sum_0^{k-1}\frac{(\lambda t)^n}{n!} - \lambda \sum_{n=0}^{k-2} \frac{(\lambda t)^n}{n!}\right)$$
Hint: $$ \frac{d}{dt} \sum_{n=0}^{k-1} f_n(t) = \frac{d}{dt} (f_0(t) + \cdots + f_{k-1}(t)) = \frac{df_0}{dt} + \cdots + \frac{df_{k-1}}{dt} =\sum_{n=0}^{k-1} \frac{df_n}{dt} $$ Also, note (after going through the product rule) that $$ \sum(f_n(t) + g_n(t)) = \sum f_n(t) + \sum g_n(t) $$