Saturday, 2 September 2017

limits - Find limntoinftyfrac(1)nnfrac325en





Question: Find limn(1)nn325en




My attempt: Since (1)n can be written as cos(πn) we can use the squeeze theorem as such: 1cos(πn)1n325encos(πn)n325enn325en and so we can take the limit as n approaches zero and because exponentials grow faster than polynomials we get: 0cos(πn)n325en0 so limn(1)nn325en=0.



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


Answer



alternative method:-|an|=n325en. Consider the series n=0|an|. Use cauchy's nth root test, we get n=0|an| converges. so limn|an|=0. Hence, limnan=0.



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