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

$ \displaystyle \sum_{n = 1}^{\infty} \frac {10^n}{(n + 1)4^{2n +1}} $

Absolutely convergent

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Missouri State University

Campbell University

Harvey Mudd College

Boston College

let's use the ratio test to determine whether the Siri's commercialise. So if we let this just let's don't this by an then the ratios has requires us to look at the limit and goes to infinity absolute value and plus one over. And so let's go ahead and write that out carefully. So we'LL have our numerator is ten to the unpleasant one and then and plus so we'LL have n plus one here instead of end and then we still add one to that because of this one over here. And then we'll have four to replace and with n Plus One and I'd still adding that other one over here because of that one. So this everything we've done so far is just our numerator and plus one. And now we'LL divide that by and which is just our formula in the beginning. So before we cancel, let's just go ahead and rewrite this we'LL flip the blue fraction and then we'Ll just multiply it to the numerator and you can see here that all the terms air positive because and is bigger than or equal to one so you can drop the absolute values here and then good and simplify those exponents And now we'LL start canceling what we can We see that we could cancel the denominator They're the ten to the end with this enough here so that'LL just give us a ten and the numerator and then looking at these terms over here If we cancel the two n plus One, we'LL just have two in the exponents left over and the denominator and then we still have and plus one over n Plus two. And as we take the limit of this expression here, that limit just approaches one. You, khun, do love his house roll if you need to to see that this limit goes toe one. So when we take the limit, we get ten over sixteen times one, which is five over eight, and that's less than one. So we conclude that the Siri's convergence by the ratio test by and then the ratio test and that's our final answer