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Numerade Educator

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Problem 10 Easy Difficulty

Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} n^{2}e^{-s^3} $

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Converges

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Video Transcript

so one way that we have often determine if this Siri's converges or diverges is through the integral test. There's one really a useful tool of intervals. So to do this, Um, what we're going to dio is take the integral from one to infinity, and with that we know that it's going to be the integral of FX, which in this case is X squared over e to the X cubed. Um And since this is how we can use the interval test because the denominator is going to increase faster than the numerator So we're going to evaluate this at VFX. So when we use substitution method, we can let x cubed equal you and three x squared d x equal, do you that allows us toe evaluate this much easier. So we ultimately end up getting is 1/3 e as our final answer and we see that these will match up. So based on this, we see that the integral converges on. Therefore, that means that the Siri's will converge