N2017 P1 Q9

A sequence of numbers \({{u}_{1}}\), \({{u}_{2}}\), \({{u}_{3}}\), … has a sum \({{S}_{n}}\) where \({{S}_{n}}=\sum\limits_{r=1}^{n}{{{u}_{r}}}\). It is given that \({{S}_{n}}=A{{n}^{2}}+Bn\), where \(A\) and \(B\) are non-zero constants.

Show that \({{r}^{2}}{{(r+1)}^{2}}-{{(r-1)}^{2}}{{r}^{2}}=k{{r}^{3}}\), where \(k\) is a constant to be determined. Use this result to find a simplified expression for \(\sum\limits_{r=1}^{n}{{{r}^{3}}}\).[4]

D’Alembert’s ratio test states that a series of the form \(\sum\limits_{r=0}^{\infty }{{{a}_{r}}}\) converges when \(\mathop{\lim}\limits_{n\to \infty }\,\left| \frac{{{a}_{n+1}}}{{{a}_{n}}} \right|<1\), and diverges when \(\mathop{\lim}\limits_{n\to \infty }\,\left| \frac{{{a}_{n+1}}}{{{a}_{n}}} \right|>1\). When \(\mathop{\lim}\limits_{n\to \infty }\,\left| \frac{{{a}_{n+1}}}{{{a}_{n}}} \right|=1\), test is inconclusive. Using the test, explain why the series \(\sum\limits_{r=0}^{\infty }{\frac{{{x}^{r}}}{r!}}\)converges for all real values of \(x\) and state the sum to infinity of this series, in terms of \(x\).[4]