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JH
Numerade Educator

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Problem 30 Easy Difficulty

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{k = 1}^{\infty} \frac {k^2}{k^2 - 2k + 5} $

Answer

The series diverges
Hint : Use divergence test

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Video Transcript

Let's go ahead and determine whether the syriza's commercial, our diversion and if it's conversion will go ahead and find that some as well. But that's a big if we'LL see. So one option here would be to try to factor into maybe do partial fractions, perhaps telescoping. But this is what the book uses. But before I go that far that will take a bit of work. I would try the diversions test. This is also weren't this way by the author, So Test four divergence. So to use this test here, this is usually easier, easier test, so should often be done first. So it's a way to use it is you just take the limit as Kay goes to infinity of this term over here. Solis, Right? Since the limit is Kay goes to Infinity case, where K squared minus two K plus five. If you can't see what this answer is, you can go ahead and let's even Let's divide a top and bottom of this bye case where? So we really haven't changed the problem because we divided top and bottom by the same thing. Then we have one over one minus two over K and then plus five over K square. And as you look, eh, go to infinity thes to terms. They're going to go to zero and you just have one over one in the limit. So our limit equals one. Therefore, let me take a step back. Since the limit equals one and this is not equal to zero, we'LL have to say the final. It's over here. The Siri's is diversion. Bye, the test for diversions and that's your final answer.