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JH
Numerade Educator

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Problem 14 Easy Difficulty

Use the Ratio Test to determine whether the series is convergent or divergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {n1}{100^n} $

Answer

Diverges

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Video Transcript

you re like tio use the ratio test to determine whether the Siri's conversions air diversions. So this means that we should look at the limit and goes to infinity absolute value and plus one over an where Anne is given by this term here. So let's just go ahead and replace the numerator and denominator. So in this case, we replace and within plus one in our formula here. So that will give you and then in the denominator and which is just this. And I removed that slew value because all these terms were positive, so there's no need for it now. Let's just go ahead and take that blue fraction, flip it upside down and then multiply to the red fraction. So now to make this would be easier. If you look at the one hundred terms here, the corresponding terms weaken right. That is one hundred to the end, and then the denominator. We can write that it is one hundred times one hundred to the end, and we could cancel those off. That will give you just the one over one hundred. And now we have to deal with these factorial terms here. So if you see and plus one fact, Auriol factorial Just go to the definition. This is just a product of all the numbers all the way up to N plus one imagers in the denominator. And so we see, we only cancel the first in terms here, and we're still left with the n plus one. Okay, so when we take that limit will have something approaching infinity divided by hundred. So this will diverge off to infinity. Now, since we're using the ratio test, we see that we have an expression that's strictly bigger than one. And that implies that the Siri's diverges yeah, and urges by the racial cyst.