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JH
Numerade Educator

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Problem 25 Easy Difficulty

Use the Root Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \left ( \frac {n^2 + 1}{2n^2 + 1} \right)^n $

Answer

Absolutely Convergent

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Video Transcript

Let's use the roots has to determine whether the Siri's conversions or diverges. So let's go ahead and call this term am The root test requires that we look at the limit and goes to infinity, and then we take the end through of a N and let's see what this value is. So go ahead and replace and with our formula. And then instead of writing the end through I'LL use what sometimes is more convenient, rational explanations. So this is our a N and then now we'LL instead of writing the radical using this idea over here, we'LL just write this as a one over and power and we know that from our laws of exponents, if you have a next opponent, be and you raise that a b a to the P power to another exponents, you can go out and just multiply those two exponents of a So, in our case, when we won't supply the exponents, they just cancel each other out and leave us with one and we're left over with and square plus one two n squared, plus one. Now, if that helps, you can go ahead and replace and with X and use local house rule here. In either case, you see that this limit is one half this number is less than one. So the Siri's converges, and that's by the root test, and that's our final answer.