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JH
Numerade Educator

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Problem 40 Easy Difficulty

A series $ \sum a_n $ is defined by the equations

$ a_1 = 1 $ $ a_{n+1} = \frac {2 + \cos n}{\sqrt{n}} a_n $

Determine whether $ \sum a_n $ converges or diverges.

Answer

converges

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Video Transcript

here were given a sequence whose first term is one and for end bigger than or equal to one earned. Let's say, two. The end term is given by this expression here, and we'd like to know whether or not the Siri's a end conversions. So let's apply on this one. Let's go for the ratio test. So that requires us to look at and plus one over n an absolute value. No. So using our formula here, we can write the numerator, and then we still have our hand on the denominator. And fortunately, those will cancel, giving us two plus co sign and over Rouen. And we can drop the absolute value because all the terms to plus coastlines positive also because co signs between negative one and one that implies two plus co signs between one and three. And so when we take the limit is n goes to infinity. The numerator will be varying between one and three, but the denominator goes to infinity so that women will be zero. That's less than one. So that means the Siri's converges by the ratio test. Okay,