E^x is less than 1/(1-x) for nonzero real x less than 1

Theorem

The following formula holds for nonzero $x \in \mathbb{R}$ with $x<1$: $$e^x < \dfrac{1}{1-x},$$ where $e^x$ denotes the exponential function.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.2.31