(a) Show that $(2 r+1)^3-(2 r-1)^3=12 k r^2+k$, where $k$ is a constant to be determined. Use this result to find $\sum_{r=1}^n r^2$, giving your answer in the form $p n(q n+1)(2 q n+1)$ where $p$ and $q$ are constants to be determined.


(b) Raabe’s test states that a series of positive terms of the form $\sum_{r=1}^{\infty} a_r$ converges when

and diverges when


the test is inconclusive. Using the test, explain why the series $\sum_{r=1}^{\infty} \frac{1}{r^3}$ converges.