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Thursday, 3 March 2016

Which one is greater - e^{\pi} or \pi^{e}?

Recall that: e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots

Therefore, \forall x > 0 , e^x > 1 + x .

Substituting x = \frac{\pi}{e} - 1 in the above inequality, we get:

e^{\frac{\pi}{e} - 1} > \frac{\pi}{e}  \implies e^{\frac{\pi}{e}} > \pi  \implies e^{\pi} > \pi^{e}

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