Series question:

How do you simplify

$$\sum_{k=0}^{\inf}2(k+2)(k+1)c _{k+2}x ^{k}+\sum_{k=1}^{\inf}k c_{k}x ^{k}+\sum_{k=0}^{\inf}c _{k}x ^{k} $$

to

4c _{2} + c _{0} + sum_{k=1}^{inf}[2(k+2)(k+1)c _{k+2}+kc _{k}+c _{k}]x ^{k}

I can't see how you would get 4c2 and c0. I know we factored out x^0 so we could get all the series to start at the same but 4c2 and c0 seems.... too much to me.

Question

Series question:

How do you simplify 

$$\sum_{k=0}^{\inf}2(k+2)(k+1)c _{k+2}x ^{k}+\sum_{k=1}^{\inf}k c_{k}x ^{k}+\sum_{k=0}^{\inf}c _{k}x ^{k} $$

to

4c _{2} + c _{0} + sum_{k=1}^{inf}[2(k+2)(k+1)c _{k+2}+kc _{k}+c _{k}]x ^{k}

I can't see how you would get 4c2 and c0. I know we factored out x^0 so we could get all the series to start at the same but 4c2 and c0 seems.... too much to me.

anonymous · Answer

$$4c _{2} + c _{0} + \sum_{k=1}^{\inf}[2(k+2)(k+1)c _{k+2}+kc _{k}+c _{k}]x ^{k}$$

anonymous · Answer

second equation.