NJC Sigma Notation Tutorial Q2
(i)
Prove that for all positive integers $n$, $\frac{1}{n!}-\frac{1}{\left( n+1 \right)!}=\frac{1}{n!+\left( n-1 \right)!}$.
(ii)
Hence evaluate $\sum\limits_{n=1}^{N}{\frac{1}{n!+\left( n-1 \right)!}}$ in terms of $N$.
(iii)
By considering $n!+n!\ge n!+\left( n-1 \right)!$ for $n\ge 1$, deduce that $\sum\limits_{n=1}^{N}{\frac{1}{n!}<2}$ for all values of $N$.
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