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

Question

Show that $\frac{1}{r!}-\frac{1}{\left( r+1 \right)!}=\frac{r}{\left( r+1 \right)!}$.[1] Find ${{S}_{n}}$ in terms of $n$, where ${{S}_{n}}=\sum\limits_{r=1}^{n}{\frac{2r}{\left( r+1 \right)!}}$.[2] Give a reason why the series converges, and write down the values of the sum to infinity.[2] Using the result in part (ii), deduce $\sum\limits_{r=3}^{n}{\frac{2r}{{{\left( r+1 \right)}^{r+1}}}}<\frac{1}{3}$.[3]

Accepted Answer

(b)(i) Shown; (b)(ii) S_n = 2(1 − 1/(n+1)!); (b)(iii) S∞ = 2; (b)(iv) Shown

2016 MI P1 Q12

2016 MI P1 Q12

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