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Why is it that I cannot use the limit comparison test for the series from n=4 to infinity of (n^2)/(n^3-3)? When I use the direct comparison test it works, but for the limit comparison test it does not work. When I do the direct comparison test, I get the Series diverges, however when I do the limit comparison test, the series converges at 1. Are these results correct or am I doing something wrong?

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1 Answer

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You are likely using the limit comparison test wrong.

Note here that$$ \lim_n \frac{\frac{n^2}{n^3-3}}{\frac{1}{n}}=1 $$Therefore, as $\sum_{n=4}^\infty \frac{1}{n}$ diverges, your series diverges.

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