Problem 3 Medium Difficulty

Use the Integral Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} n^{-3} $

Answer

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Video Transcript

to do this problem, We need to know that we have a continuous, positive, decreasing function and the function f f X equals X to the negative three. Or, in other words, one over X Cubed fits his criteria. It's graph looks something like this you're wondering. So now we can it say that the sum from one to infinity of end of the native three is less than any girlfriend. Want to infinity of one over X cubed dicks? So we shall now do is evaluate this integral and see if it's convergence. If it is convergent, then we have the seriousness conversion. So let's take a look. Since this is an improper integral, we're going to need to take a limit. And this is actually not too bad of an integral as funerals go, Ah, anti derivative, you know, in this year's nice power rule and we can we can already see where this is starting the head because X to the negative too is one over x squared. So this becomes negative. One half times one over t squared plus one half and we're taking the limit. Of course, as he goes to infinity and Steve goes to infinity. This term here goes to zero. So we're left with just one half. So the girl is convergent. And since the Siri's is less than the inner girl, we know Siri's itself is conversion.