Suppose that $\{\epsilon _1 , · · · , \epsilon_m \} $are real numbers that all satisfy $|\epsilon_i | ≤ \eta$. Show that given $C > 1$, we have that $$ \Pi_{j=1}^m(1+\epsilon_j) = 1 + \epsilon$$where $|\epsilon| ≤ Cm\eta$, provided that $0 < \eta ≤ \min\{ log \frac{C}{m-1}, 1\}$. For double precision arithmetic how large can we take $m$ if $C = 2$?
2026-02-23 01:01:53.1771808513
Bounds on real numbers
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