Difference between revisions of "E^x greater than (1+x/y)^y greater than exp(xy/(x+y) for x greater than 0 and y greater than 0)"
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(Created page with "==Theorem== The following formula holds for $x>0$ and $y>0$: $$\exp \left( x \right) > \left( 1 + \dfrac{x}{y} \right)^y > \exp \left( \dfrac{xy}{x+y} \right),$$ where $\exp$...") |
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==References== | ==References== | ||
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=E^x greater than 1+x^n/n! for n greater than 0 and nonzero real x greater than 0|next=e^(-x) less than 1-(x/2) for 0 less than x less than or equal to 1.5936}}: 4.2.36 | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=E^x greater than 1+x^n/n! for n greater than 0 and nonzero real x greater than 0|next=e^(-x) less than 1-(x/2) for 0 less than x less than or equal to 1.5936}}: 4.2.36 | ||
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Revision as of 20:16, 7 June 2016
Theorem
The following formula holds for $x>0$ and $y>0$: $$\exp \left( x \right) > \left( 1 + \dfrac{x}{y} \right)^y > \exp \left( \dfrac{xy}{x+y} \right),$$ where $\exp$ denotes the exponential.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.2.36
