Prove that for all positive integers $n$, $\frac{1}{n!}-\frac{1}{\left( n+1 \right)!}=\frac{1}{n!+\l

Question

Prove that for all positive integers $n$, $\frac{1}{n!}-\frac{1}{\left( n+1 \right)!}=\frac{1}{n!+\left( n-1 \right)!}$. Hence evaluate $\sum\limits_{n=1}^{N}{\frac{1}{n!+\left( n-1 \right)!}}$ in terms of $N$. By considering $n!+n!\ge n!+\left( n-1 \right)!$ for $n\ge 1$, deduce that $\sum\limits_{n=1}^{N}{\frac{1}{n!}

Accepted Answer

(i) shown; (ii) $1-\frac1{(N+1)!}$; (iii) $\displaystyle \sum_{n=1}^{N}\frac1{n!}<2$

NJC Sigma Notation Tutorial Q2

NJC Sigma Notation Tutorial Q2

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