Question: Find limn→∞(−1)nn325en
My attempt: Since (−1)n can be written as cos(πn) we can use the squeeze theorem as such: −1≤cos(πn)≤1⟹−n325en≤cos(πn)⋅n325en≤n325en and so we can take the limit as n approaches zero and because exponentials grow faster than polynomials we get: 0≤cos(πn)⋅n325en≤0 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, limn→∞an=0.
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