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Problem 11 Easy Difficulty
Use the Ratio Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{k = 1}^{\infty} \frac {1}{k!} $
Answer
absolute convergence is the same as convergence.
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Video Transcript
let's use the ratio test to determine whether the Siri's converges air averages. Now the ratio test requires that we look at this term a k one over k factorial. So we'd like to look at the limit as Kay goes to infinity. Absolute value, a k a. Plus one over a. K. So let's go ahead and write off this fraction in the numerator. That's one over Kaye plus one factorial have all in the denominator there. We just see a K. We already know what that is, so that goes down here in the denominator. And we couldn't drop the absolute value because the fractions are These are all positive numbers. So we have k factorial over Kaye plus one factorial. Let's go ahead and rewrite that denominator as que factorial times K plus one. Cross off those cave editorials and we just have limit one over Kaye plus one as Que Goes to infinity dis limit become zero. That's less than one. So we conclude at the original Siri's to sum from one to infinity converges by the ratio test, and that's our final answer
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