guys, can you apply differentiation to normal summation series? like below

Question

turingtest · Answer

You can differentiate or integrate series pretty much normally.

anonymous · Answer

$$f(x)=\sum_{n=1}^{K}(a)^n$$ then $$f \prime (x )=n \sum_{n=1}^{K}(a)^{n-1}$$ ?

anonymous · Answer

is it so?

turingtest · Answer

no, because you've got f(x) and yet you have n's and a's in the problem
what are you differentiating with respect to? n ?

anonymous · Answer

opps it is f(a) not f(x), sorry for the typo

anonymous · Answer

differentiate with respect to a.

turingtest · Answer

in that case it's right

kc_kennylau · Answer

except that the n should be put inside the summation

anonymous · Answer

thank you so much turing, i never knew that differentiation can be applied to summation

turingtest · Answer

^yeah that too
if you are in doubt, just imagine the series expanded$$f(a)=a+a^2+a^3+\cdots$$$$f'(a)=1+2a+3a^2+\cdots$$differntiating term by term you can see it turns out as @kc_kennylau said

kc_kennylau · Answer

$$\Large\frac d{dx}\sum_{n=1}^Kx^n=\sum_{n=1}^Knx^{n-1}$$