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Determine whether the series converges or diverges.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {4^{n + 1}}{3^n - 2} $

diverges

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Oregon State University

Baylor University

University of Michigan - Ann Arbor

determine whether this given series converges or diverges Here. I'll use the comparison test or direct comparison. The book might call it so this one does not require a limit. So here I'll just compare this some to another some. So first I'll use the fact that we have four n plus one over three to the N minus two. This is bigger than or equal to. Really. It's just strictly bigger than three to the end. And the reason for this? The reason the left side is bigger is because it's the nominator smaller. So therefore, yeah, the terms that were adding or smaller. This means that the entire sum is smaller. So I couldn't go ahead and replace the some with the smaller Some here because I'm using the inequality. And then I can rewrite this as a geometric series if I pull out the four. So here, just take this four out. So we have four to the end over three at the end, so I'll just write. That is four times for over three to the end power, and now we can see that this is geometric and we could even see that they are the value that we're continuing to multiply by each time we increase in. It's 4/3, however, are they satisfies this. The absolute value of our is bigger than one in any time. The absolute value of our is bigger than or equal to one. The series diverges. This is not our series. This is the geometric series. However, now we use the comparison test we have that our series is larger than the divergent series. So by the comparison test, our series also diverges bye, and I'll just abbreviate this by C t for a comparison test. Our series will also convert or also excuse me. Our series will also diverge because the series, the smaller series diverged and that's our final answer.