List of representations of e: Difference between revisions

→‎As an infinite series: minor simplification: 3 - \sum_{k=2}^\infty \frac{1}{k! (k-1) k}
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(→‎As an infinite series: minor simplification: 3 - \sum_{k=2}^\infty \frac{1}{k! (k-1) k})
 
Consideration of how to put upper bounds on ''e'' leads to this descending series:
:<math>e = 3 +- \sum_{k=2}^\infty \frac{-1}{k! (k-1) k} = 3 - \frac{1}{4} - \frac{1}{36} - \frac{1}{288} - \frac{1}{2400} - \frac{1}{21600} - \frac{1}{211680} - \frac{1}{2257920} - \cdots </math>
which gives at least one correct (or rounded up) digit per term. That is, if 1 ≤ ''n'', then
:<math>e < 3 +- \sum_{k=2}^n \frac{-1}{k! (k-1) k} < e + 0.6 \cdot 10^{1-n} \,.</math>
More generally, if ''x'' is not in {2, 3, 4, 5, ...}, then
:<math>e^x = \frac{2+x}{2-x} + \sum_{k=2}^\infty \frac{- x^{k+1}}{k! (k-x) (k+1-x)} \,.</math>