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JH
Numerade Educator

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Problem 15 Easy Difficulty

Let $ a_n = \frac {2n}{3n + 1} $
(a) Determine whether $ \left\{ a_n \right\} $ is convergent.
(b) Determine whether $ \sum_{n = 1}^{\infty} a_n $ is convergent.

Answer

(A). The sequence converges to $\frac{2}{3}$
(B). The series diverges

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Video Transcript

So our sequence is given by the formula to a number three and plus one sulfur party. Eh? We'd like to know whether or not this converges. So we just have to check. Is this thing a real number? That's basically what they're asking. So the limit has to exist, but it can't be plus or minus infinity. So let's go out and compute. That limit is angles to infinity to end over three and plus one. So let's go to the denominator. I see that the highest power of and there is one. So let me go ahead and just divide top and bottom by And so another me, right, it's in a nicer way. So I'll go ahead into by the top I end. But then that will cancel out because I'm doing it again on the denominator. So I get the limit and goes to infinity of two over three plus one over end and you know, by your limit laws, you could go ahead and take this limit here and distributed toe all the terms. And when we distributed to one over end that'LL go to zero in the limit, the whole fraction will. So we're just left with two or three plus zero, which is to over three. So that's our answer for party. That's the limit of the sequence. Now we're going onto the Siri's. So be it. Here's our answer. So before I write the answer, let me take a step back. We'LL call your test for diversions. This is a key test for to use for Siri's, The test says. If so, limit of a end does not exist or if the women does exist. But it's not a but it's not equal to zero. Then the Siri's will diverge. So on our problem, let me write this in a different color. Bye. Here's our answer by the test for divergence, since in party we showed that the limit of a hen and goes to infinity was equal to two or three. So it exists, but it's not equal to zero. The Siri's, which is given up here, diverges, and that's our final answer.