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Numerade Educator

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Problem 5 Medium Difficulty

Use the Integral Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {2}{5n - 1} $

Answer

The given series diverges

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Video Transcript

So we have here a positive continues decreasing function, which means we can use the integral test. So we'LL take the integral from one to infinity to over five X minus one t x And of course, we need to change that into a limit and we're gonna use you substitution to do this integral. So we'll use the substitution with u equals five x minus one, which would make do you five d x. So this is really equal to the limit The studio's turn affinity any girl from one to infinity Or really, because we're changing the bounds, we should say so. Kind of a to b of too over you times to you over five The do over five comes from dividing the five from r D X over to the d you side. Now we can integrate giving us two fifths Allen of the absolute value of you. Ah, for the bounds of our into girls and replacing our ex right peace back in for you. We have this for integral. Now when we actually plug our bounds of integration, would that be should be a t We find that we end up with I two fifths, Alan five T minus one minus two fifths. Fell into four. No. And as t goes to infinity, this whole thing goes to infinity, so we know the Siri's is diversion.