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JH
Numerade Educator

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Problem 39 Easy Difficulty

The terms of a series are defined recursively by the equations

$ a_1 = 2 $ $ a_{n+1} = \frac {5n + 1}{4n + 3} a_n $

Determine whether $ \sum a_n $ converges or diverges.

Answer

The series diverges.

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Video Transcript

given the sequence, and the first term is, too. And the following terms were given by the second equation here, and we'd like to know whether the sum of an conversions or diverges. So to do that, let's apply the ratio test, which requires that we look at the limit. So we're about the limit later, but the fraction is and plus one over and now, using our formula Pierre, we can write the numerator as such, and the denominator is just an now We could cancel those and terms, and we're just left with five plus one over foreign plus Tree I'LL drop the absolute values here because the remaining terms are all positive. Now we take the limit, and this let me goes to If you ply, let's say example low Patel's rule or some algebra you get five over for, which is bigger than one. So for this, we conclude that the Siri's A M diverges by the ratio test, and again, the reason we're saying diverges instead of converges is because the absolute value of and plus one over N and the limit is Shrink Lee larger than one