1+x greater than exp(x/(1+x)) for nonzero real x greater than -1

Theorem

The following formula holds for nonzero $x \in \mathbb{R}$ with $x>-1$: $$1+x > \exp \left( \dfrac{x}{1+x} \right),$$ where $\exp$ denotes the exponential.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.2.34$