Tuesday, October 31, 2017

real analysis - limlimitsnto+inftysqrt[n]n! is infinite




How do I prove that lim is infinite?


Answer



By considering Taylor series, \displaystyle e^x \geq \frac{x^n}{n!} for all x\geq 0, and n\in \mathbb{N}. In particular, for x=n this yields n! \geq \left( \frac{n}{e} \right)^n .



Thus \sqrt[n]{n!} \geq \frac{n}{e} \to \infty.


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