2023 SAJC P2 Q5

Show that \(\frac{1}{r!}-\frac{1}{\left( r+1 \right)!}=\frac{r}{\left( r+1 \right)!}\).[1]

Using the result in (i), evaluate the sum \({{S}_{n}}=\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...+\frac{n}{\left( n+1 \right)!}\) in terms of \(n\).[3]

Explain why the series converges as \(n\to \infty \) and state its sum to infinity \({{S}_{\infty }}\).[2]

The Comparison Test for Convergence states that if \(\sum\limits_{r=1}^{\infty }{{{a}_{r}}}\) and \(\sum\limits_{r=1}^{\infty }{{{b}_{r}}}\) are two infinite series with \({{a}_{r}}\ge 0\) and \({{b}_{r}}\ge 0\) and \({{a}_{r}}\le {{b}_{r}}\) for all \(r\in {{\mathbb{Z}}^{+}}\), then \(\sum\limits_{r=1}^{\infty }{{{b}_{r}}}\) converges implies that \(\sum\limits_{r=1}^{\infty }{{{a}_{r}}}\) converges.

Using the Comparison Test for Convergence and result from part (iii), explain why the series \(\sum\limits_{r=1}^{\infty }{\frac{{{\left( r-1 \right)}^{2}}}{\left( r+2 \right)!}}\) converges.[3]