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Problem 8 Easy Difficulty

Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} ( - 1)^{n-1} \frac {n^4}{4^n} $

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Converges

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Video Transcript

two things toe observe here is that we're alternating sign and that we have polynomial growth on top and exponential growth at the bottom. So the alternating signed tests. We need to have that. Our terms are eventually strictly decreasing in absolute value to zero, so they can't be oscillating and then approaching zero, they need to be strictly decreasing to zero in absolute value. But since exponential growth is eventually going to be faster than polynomial growth, we will have eventually that the absolute value of how the terms in consideration will be strictly decreased into zero, which is what we want. So let me just write this down. Okay? So this is one of the things that we wanted to happen. We wanted for this limit, be zero. Okay, so that's important. And as we mentioned, it's also important that, you know, this is these are all positive. Thanks. Right, because we're just looking at the absolute value of this. So we want the absolute value to be decreasing zero like we're accomplishing here. So if we get this to happen and we're alternating signs than the alternating signed test gives us a convergence, Okay, so you should know that this is going to be a limit that goes to zero. Um, yeah, basically, because, yeah, this is polynomial, and this is exponential girls. But if you really want to be rigorous about it, you could always ah, apply Lopa towels rule, and then you do the derivative. Or if you do, Lopata was rule four times. Then the numerator is just goingto turn in tow some constant. And then we'll still have something that blows up to infinity and the denominator When you apply lope it'll four times and then you you'Ll see that this limit is indeed zero. But it should be intuitively obvious that exponential growth is faster enough that we're goingto accomplish this limit being zero.