Solution
= Solution
The two finite <summations> cancel term by term after shifting the index. Only the final positive term remains:
$$
\boxed{\sum_{n=0}^N(n+1)^3-\sum_{n=0}^Nn^3=(N+1)^3.}
$$
This is a <telescoping sum>.
= Solution
The two finite <summations> cancel term by term after shifting the index. Only the final positive term remains:
$$
\boxed{\sum_{n=0}^N(n+1)^3-\sum_{n=0}^Nn^3=(N+1)^3.}
$$
This is a <telescoping sum>.