Saturday, 2 September 2017

limits - Find limntoinftyfrac(1)nnfrac325en





Question: Find lim




My attempt: Since (-1)^n can be written as \cos(\pi n) we can use the squeeze theorem as such: -1 \leq \cos(\pi n) \leq 1 \implies \frac{-n^{\frac{3}{2}}}{5e^n} \leq \frac{\cos(\pi n)\cdot n^{\frac{3}{2}}}{5e^n}\leq \frac{n^{\frac{3}{2}}}{5e^n} and so we can take the limit as n approaches zero and because exponentials grow faster than polynomials we get: 0\leq \frac{\cos(\pi n)\cdot n^{\frac{3}{2}}}{5e^n}\leq 0 so \lim_{n \to \infty} \frac{(-1)^n n^{\frac{3}{2}}}{5e^n} =0 .



Is this correct? If so what other ways could I have solved this limit?


Answer



alternative method:-|a_n|=\frac{n^{\frac{3}{2}}}{5e^n}. Consider the series \sum_{n=0}^{\infty}|a_n|. Use cauchy's nth root test, we get \sum_{n=0}^{\infty}|a_n| converges. so \lim_{n\to\infty}|a_n|=0. Hence, \lim_{n\to\infty}a_n=0.



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