I am reading this note on the Jackknife estimator, and I'm confused about this part:
Suppose $$ \sum ^\infty _{k=1} \frac{a_k}{n^k}=O(\frac 1 n), $$ then $$ \sum ^\infty _{k=1} a_k (\frac 1 {n^{k-1}}-\frac{1}{(n-1)^{k-1}}) = \frac{a_2}{n^2} + O(\frac 1{n^3}) $$
Why is this?
I know the definition of the big-O notation but that's pretty much all I know about it so a thorough explanation would be much appreciated. It would be nice too if someone could tell me where to read more about it. Thank you!