Interchange of differentiation and summation

Question

I came across an example about interchange of differentiation and summation. Can anyone show me how to prove the equation in the picture? Thank you!

Answer 1

You do not interchange differentiation and summation. Obviuosly $\sum_{x=0}^n \theta x (1 - \theta)^{x-1} = \theta \sum_{x=0}^n x (1 - \theta)^{x-1}$. In the second step you use that $x (1 - \theta)^{x-1} = -\frac{\partial}{\partial \theta} (1 - \theta)^x$ for each summand.