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Numerade Educator

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Problem 12 Easy Difficulty

Test the series for convergence or divergence.

$ \displaystyle \sum_{k = 1}^{\infty} \frac {1}{k \sqrt{k^2 +1}} $

Answer

converges

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Video Transcript

to this problem. It's important to keep in mind that when you make the denominators smaller, you end up with something that's bigger. Okay, so if we change the case squared plus one to just case squared would be making the denominator smaller, so we would be getting something that's even bigger than this guy. Okay, but the square root of case squared that's absolute value of K. And if our case, they're all positive, then it's just regular K. Useless can again be simplified toe one over Hey, squared. And this is something that's finite, Kate. And then it's important to notice that all of these terms are positive. Okay, so if we're if our terms, they're all positive and we're finite, then we have convergent. We have convergence, positive and finite, positive terms and finite implies convergence. Okay, so you can come up with some examples of where things are alternating signs and you're you. You might be bounded, but ifyou're alternating signs, then being bounded is not always good. Enoughto get convergence. Hey, but if all of your terms are positive and your bounded, then you get convergence and that's that's what's happening here