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[itex]\frac{\partial}{\partial r}[/itex](r[itex]\frac{\partial T}{\partial r}[/itex])+[itex]\frac{\partial ^{2}T}{\partial\Theta^{2}}[/itex]=0

(2): [itex]\frac{T-T_{0}}{T_{0}}[/itex]=A[itex]_{0}[/itex]+[itex]\sum[/itex] from n=1 to infinity of ([itex]\frac{r}{R}[/itex])[itex]^{n}[/itex](A[itex]_{n}[/itex]cos(n[itex]\Theta[/itex])+B[itex]_{n}[/itex]sin(n[itex]\Theta[/itex]))

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