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

$ \displaystyle \sum_{n = 1}^{\infty} \frac {n^{10}}{( - 10)^{n+1}} $

Absolutely Converges

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Missouri State University

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let's use the ratio test to determine whether the Siri's conversions or divergence to do that, We should look at this term Over here, this's our end. So let's end to the tenth Power Overnegative ten to the end, plus one power. And the ratio test involves this limit of the absolute value and plus one over. And and the answer to this limit will tell us whether the Siri's Khun may tell us whether the series conversions of laboratories now in the numerator Let's do that and read So we'LL have n plus one to the ten negative ten and Plus two still have the slum it over here as and goes to infinity for the denominator. Let's do that in Green and we just use our formula over here. Now let's go ahead and rewrite this before we take the limit we have and plus one to the ten and to the ten. And then here we could drop the absolute value. Just make sure you replace these minus signs, so we'LL have ten to the N plus one over ten to the and plus two. So for this fraction on the right, most fraction you could cancel all the tens up there and then you'LL have one left in the bottom. And also, if you take the limit of this expression over here, you can also write. This is and plus one over into the tent, and a limit of this is one to the ten, which is one. So we have one times one over his head. That's one over ten, which is less than one and therefore the Siri's converges by the ratio test, and that's our final answer.