Help With Partial Derivatives and Infinite Sums

[Tenenbaum3r]

I'm working on a calculus project and I can't seem to work through this next part...
I need to substitute equation (2) into equation (1):

(1): r\frac{\partial}{\partial r}(r\frac{\partial T}{\partial r})+\frac{\partial ^{2}T}{\partial\Theta^{2}}=0

(2): \frac{T-T_{0}}{T_{0}}=A_{0}+\sum from n=1 to infinity of (\frac{r}{R})^{n}(A_{n}cos(n\Theta)+B_{n}sin(n\Theta))

I know I have to solve for T in the second equation and then substitute but I don't really know the rules for infinite sums... The whole point of this is to prove that equation (2) is a solution to equation (1). Any help or advice would be appreciated!

 

[mfb]

You can multiply an infinite sum with T₀, this is no problem. You don't need to modify the sum itself to solve equation (2) for T.

 

[Tenenbaum3r]

Thank you! that helped me figure it out