Problem 13 Medium Difficulty

Determine whether the series is convergent or divergent.
$ \frac {1}{3} + \frac {1}{7} + \frac {1}{11} + \frac {1}{15} + \frac {1}{19} + \cdot \cdot \cdot $

Answer

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Video Transcript

since this is a continuous positive, decreasing function on the interval from one to infinity, we will use the integral test realizing their Siri's is one over to end plus one. So we will evaluate the integral from one to infinity of one over two X plus one suspect x, which means we have to take a limit. Uh, t ghost Infinity from wonder t of that function. And to integrate this we'LL do a u substitution Ah, where we will just have you equal to two x plus one which makes our do equal to two d x or in other words, one half to you is equal to the ex. So when we make our substitution, get the limit as T goes to infinity groom from some some coordinates here Andi won over you do you times one half We'LL take the one half hour in the front one half Really? We take the one half into the front of the integral. However property of limits is you can take constant coefficients in front limit as well. So just do it here for simplicity And when we integrate, we end up with the natural log of you evaluated from whatever those end points are. When we replace our exes back for youse, they will get one half the limit is he goes to infinity. Ah, Ellen. Two tea plus one minus Ellen of one. No, this whole difference in so gloomy. Ah, in between. In between these two steps I replaced the you with two x plus one from before and then plugged in the tea in the one from before. And as t goes to infinity Ah, Ellen to t plus one goes to infinity. So this whole, uh hola, mate goes to infinity. So because one half limit as t to infinity of the line of to t plus one minus on the one which is really just hero, that goes to infinity. So the siri's this diversion.