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JH
Numerade Educator

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Problem 2 Easy Difficulty

Test the series for convergence or divergence.
$ \frac {2}{3} - \frac {2}{5} + \frac {2}{7} - \frac { 2}{9} + \frac {2}{11} - \cdot \cdot \cdot $

Answer

Converges

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Video Transcript

Let's see whether the Siri's conversions or diverges Now we can see that this can be written well, what's the pattern? Here? We see that the sign alternates and what's going on in the denominator. It looks like it's increasing by two denominator increases by two. So in the top, I should just have a too. In the denominator we can do to one plus one. And then because the sign alternates, I should have something like negative one. In this case, let's through to the end, plus one. Okay, so that is our Siri's. You can check if you plug in a few values event that it will match up with the Siri's over here. Yes, and is one. We get two or three. A pen is, too. We have negative two of top and then five in the bottom and so on. Now here. Let's use the alternating Siri's test. So here we should say what our BN is. That's just the bien. So we ignore the minus one to the end there, and then we have to check that the limited beyond zero. That's true. It's also we should have checked it originally that the being was positive. And that is true over here. And we also need that the end is decreasing. Yeah, so this means the end plus one is less than or equal to be in. So on our keys two over two and plus one plus one is less than two over to n plus one. And this is true because the denominator on the left is larger, so larger denominator. Therefore, the Siri's converges converges by the alternating Siri's tests. Army has called this a s t. So the Siri's converges by a and that's the final answer.