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I was working through a problem about the convergence of a power series and got stuck on the radius of convergence. My textbook says to use the ratio test, but when I apply it to the series ∑ (n² * xⁿ) / 3ⁿ, I end up with a limit involving |x|/3. I think the radius should be 3, but I'm not sure if I handled the n² term correctly in the limit calculation.

the ratio test gives a limit that looks like the absolute value of x divided by three and the n squared part drops out because the ratio of n plus one squared to n squared tends to one

when x is three the terms are n squared and do not go to zero so the series cannot converge

when x is minus three the terms also do not go to zero because you still have the magnitudes growing and the sign does not save it

another angle is to view the series as a polynomial in n times a geometric piece with ratio x divided by three so for x smaller in absolute value than three it should converge

i remember that the boundary needs a separate check and the ratio test alone is not enough for the end points

one quick way i tried was to compare with a known fact that a polynomial times a geometric series behaves like a geometric in terms of radius

could you check what happens at the ends to be sure

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