Ratio Test for following limit

Question

Evaluate the following limit.

∞ (-7)^n

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n = 1 n^2

Answer 1

Hi James!

I always find it useful to think about these problems first and make a prediction whether or not I think it should converge or not. Its a bit less useful with an alternating series but still a good practice. In this case I would think about the rate of growth of 7ⁿ versus n². The numerator in this case grows much faster than the denominator so the series will not converge. So we should expect our limit to be greater than 1, indicating that the series does not converge. Now for the actual limit to verify our hypothesis (hopefully).

lim_n→∞|(a_n+1) / a_n| = lim_n→∞|((-1)ⁿ⁺¹7ⁿ⁺¹n²) / ((-1)ⁿ(n+1)²7ⁿ)| : plug in values

= lim_n→∞|(-1)(7n²) / (n+1)²| : reduce exponential component

= lim_n→∞|(-7n²) / (n²+2n+1)| : expand denominator

= lim_n→∞|-7n²/n²| : drop terms with smaller powers

= lim_n→∞|-7| = 7 = p

So we get that the limit of the corresponding positive series is 7, which validates our prediction that p > 1. This means that the corresponding positive series MAY diverge. In order to determine this we also would take the limit of a_n (minus the (-1)ⁿ part) as it approaches infinity in order to see if the alternating series converges conditionally. In this case we can tell by inspection that the limit will be infinity since the numerator grows faster than the denominator. So the alternating series does diverge. Hope this helps!

-Nathan