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JH
Numerade Educator

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Problem 6 Easy Difficulty

Test the series for convergence or divergence.
$ \displaystyle \sum_{n = 0}^{\infty} \frac {(-1)^{n + 1}}{\sqrt {n + 1}} $

Answer

Converges

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Video Transcript

let's determine whether the Siri's conversions or diverges now the first thing that I noticed is that the syriza's alternating and that's just because of this negative one to the end power up here that will make the sign change. So here the first thing that comes to mind when I have altering series is to try the alternating Siri's test. So first, this is our end. And then we have bee n equals one over and plus one. So be it is everything in an after you take out the negative one to the end, plus one power. So here this is positive. So that's the first condition that needs to be satisfied when using this altar earnings a recessed. The second condition is we need the LTD be end to be zero and we can see that this is true. As then gets larger, the denominator goes to infinity. But the numerator is just one so that zero and lastly we need bien plus one to be less than or equal to be in. This's the decreasing condition. So here bien plus one is just one over radical and plus one, whereas bn is just one over radical and and this inequality is true, sends the denominator on the left is larger. Larger denominator means smaller fraction. So that's why we have the inequality. This satisfies all the conditions for the ulcer, any serious test. So we conclude the Siri's, which was from n equals zero to infinity. Negative one to the end, plus one over radical and plus one. This thing will converge by the alternative theories test. Yeah, and that's our final answer.