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JH
Numerade Educator

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Problem 22 Medium Difficulty

Use the Ratio Test to determine whether the series is convergent or divergent.

$ \frac {2}{3} + \frac {2 \cdot 5}{3 \cdot 5} + \frac {2 \cdot 5 \cdot 8}{3 \cdot 5 \cdot 7} + \frac {2 \cdot 5 \cdot 8 \cdot 11}{3 \cdot 5 \cdot 7 \cdot 9} + \cdot \cdot \cdot $

Answer

divergent

Discussion

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Video Transcript

to use the ratio test here. It would be helpful tohave the formula for the end term of the theories. Now we see here are first term a one a two and so on. And if we keep going on in this pattern you see in the numerator we start off with two and then we keep adding three. So it looks like we should have three and minus one. And you can check if an equals one. You too. If any calls to you get five and each time and increases the new writer and includes an additional factor of three increases by three. Now in the denominator, you see that we have three, five and so on and in the denominator or increasing by two. So this time we should have ah too, and and there instead. But when we plug in n equals one, we'd like to get three. So we should do two and plus one. So here's our formula for the INSERM. And so now we go to the ratio test This we look at the limit and goes to infinity and plus one over. And so now I could drop the absolute values because all of our numbers were positive. And now here, let's do a and plus one. So the numerator over here and then we would keep multiplying. And then we would include the next term. So that's the numerator for an plus one, Sol Do listen blue and the denominator and then we increase and buy one and I will divide buy n So we'Ll just use our formula here and now, as usual will go ahead and take that denominator flip it upside down and multiply it. So here, running out of a bit of room here. So let's see if we could squeeze this in and notice the very last term over here. If you simplify this, that's three n plus two in the denominator and then the very last term here, that's a two one plus three. No. Okay, now we're multiplying this by this thing after we flip it upside out and that will start canceling out as much as we can. So we look at the denominator over here and you see that you could cancel all the terms except the last one with this numerator over here. So that leaves us with a to N plus three. And the denominator. How about the numerator? It looks like we could cancel and then here. Sorry. That should have been a five. It looks like you could cancel everything up to three and minus one, and then you're left with to re n plus two. At this point, if it helps, you can use local tells rule toe, evaluate that limit. In either case, you get three over to which is bigger than one. So the Siri's diverges by the ratio test.