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JH
Numerade Educator

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Problem 42 Medium Difficulty

Let $ \{ {b_n} \} $ be a sequence of positive numbers that converges to $ \frac {1}{2}. $ Determine whether the given series is absolutely convergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^n n!}{n^nb_1b_2b_3 \cdot \cdot \cdot b_n} $

Answer

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Hint : Use Ratio test

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Video Transcript

here. We're given that being conversions to a half and we want to know whether the Siri's here is absolutely conversion. Yeah, So I'll go ahead and call this term here This is our end And then I used the ratio test so we'll take a limit at the end for right now licious worry about the fraction some dealing with the numerator here. So just replace the red formula with n plus one. And then now we divide by an So we're dividing by this fraction over here. So we'll flip it upside down and then multiplied in All right, Celeste from the previous part. So now we're just dealing with inside the absolute value so we can cancel some terms here. So we see we could cancel be won through bien be won through being there. That leaves us with being plus one of the bottoms and recall and plus one factorial That's n factorial times and plus one, and that will allow us to cancel those factorial is there leaving us with n plus one of the top and I should have had also a plus one here on the n plus one. Sorry about that, typo and then we still have entered the end as well. So we could cancel more here. So that's n plus one. Cancel that with one there and then I'll rewrite this as an over and plus one to the end. Now we're ready to take a limited so being converges to a half so that means one over being converges toe one over a half, which is too. So when we take the limit here also, we'LL have to deal with this fraction here. So this one's a little trickier. So notice that this is the reciprocal of one over one plus one over into the end and then by definition, if you take the limit here, that denominator goes tio good. So this fraction is we take a limit as n goes to infinity. For the ratio test, we get two divided by which is less than one. So we conclude that the Siri's is absolutely conversion by the ratios ist And that's because our limit of and plus one over and an absolute value was less than one