Difference between revisions of "E^(-x/(1-x)) is less than 1-x is less than e^(-x) for nonzero real x less than 1"
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==Theorem== | ==Theorem== | ||
| − | The following formula holds for $x \in | + | The following formula holds for nonzero $x \in \mathbb{R}$ with $x<1$: |
| − | $$\exp \left( -\dfrac{x}{1-x} \right) < 1-x < e^{-x},$$ | + | $$\exp \left( - \dfrac{x}{1-x} \right) < 1-x < e^{-x},$$ |
where $\exp$ denotes the [[exponential]]. | where $\exp$ denotes the [[exponential]]. | ||
| Line 7: | Line 7: | ||
==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=|next=}}: 4.2.29 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=e^x is greater than 1+x for nonzero real x}}: $4.2.29$ |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 00:31, 23 December 2016
Theorem
The following formula holds for nonzero $x \in \mathbb{R}$ with $x<1$: $$\exp \left( - \dfrac{x}{1-x} \right) < 1-x < e^{-x},$$ where $\exp$ denotes the exponential.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.2.29$
