use-the-integral-test-to-determine-whether-the-series-is-convergent-or-divergent-displaystyle-sum_n-

Question

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

Clayton Craig · Accepted Answer

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&#x27;s graph looks something like this you&#x27;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&#x27;s convergence. If it is convergent, then we have the seriousness conversion. So let&#x27;s take a look. Since this is an improper integral, we&#x27;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&#x27;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&#x27;re taking the limit. Of course, as he goes to infinity and Steve goes to infinity. This term here goes to zero. So we&#x27;re left with just one half. So the girl is convergent. And since the Siri&#x27;s is less than the inner girl, we know Siri&#x27;s itself is conversion.

Problem

Use the Integral Test to determine whether the se…

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

The given series converges

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