Problem 20 Medium Difficulty

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

Answer

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Video Transcript

Let's test the Siri's for conversions or diversions. Now, although it is alternating, I'm curious to see what the limit of this term will be, because if it's non zero, then it'LL diverge. So here a common trick is toe multiply, numerator and denominator, by the contrary, it So let's go ahead and multiply that out that'LL give us the limit as N goes to infinity. Come on. So in the numerator we're just left with one After you simplify war is in the denominator and plus one plus radical end it. This does go to zero when that's a good thing, so this may have some hope of commercial. So let's call. First of all, we know that this is all turning Siri's check because of this negative one to the end Power. Then, if we call Bien to just be the positive, are just absolute value of the A M. This is always positive because the first radicals bitter and that's the one of the conditions that's needing. That's needed if using the alternating Siri's test. So the second condition that's needed is that the limit of being is equal to zero. But we just showed in the first part of this problem we showed that was true. And then the final condition. Part three is that the bien the sequence is decreasing. So if we can show that the beyond sequences eventually decreasing, then we can imply that the Siri's converges by the Austrian ing seriousness. So let's go to the next page in re right at the end. Now to show this is decreasing, we could try to show that this is true. So in our case, and plus two minus radical and plus one is less than or equal to radical and plus one minus radical in now the difficulty of showing this it it could be done, possibly with some algebra. But it might be easier to show that bees decreasing by considering this continuous function. F let's rein in this form. And then we can show that if f prime is negative, that means that F is decreasing and that's equivalent to be and decreasing so we can avoid doing the tedious algebra and dealing with these radicals. Rather than doing that, we just take a derivative. So here f prime of X. So we have one half x plus one to the negative. Let me rewrite this one over to radical X plus one minus one over to radical X. And this is definitely negative because the second term that's being subtracted is larger because I had a small denominator. So this is true. So that means BN is depressing. And that was the third condition that we needed for the alternate endings. A recess. So therefore the Siri's converges Bye, the alternating Siri's cyst, and that's your final answer.