Limit comparison test, rational function (n^2-5n+1)/(4n+n^4). How to choose the limit comparison.

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In order to choose the comparison for this rational function series, we look at the highest power of n in the numerator and denominator; that is, the terms that will dominate as n becomes large. We see immediately that the terms of the series essentially settle down to 1/n^2 in the large n limit, so that's what we choose for the limit comparison. Taking the limit of the ratio of terms, we obtain a finite number and conclude that our series converges.