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!

Just differentiation: $-\frac{\partial}{\partial\theta}(1-\theta)^x=x(1-\theta)^{x-1}$ — StubbornAtom, Dec 12 '18 at 06:05
There is no prob!em with finite sums---convergence is not an issue — kjetil b halvorsen, Dec 12 '18 at 07:20

score 0 · Answer 1 · answered Dec 12 '18 at 15:38

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.

Interchange of differentiation and summation

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