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JH
Numerade Educator

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Problem 7 Easy Difficulty

Use the Ratio Test to determine whether the series is convergent or divergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {n}{5^n} $

Answer

absolutely convergent by the Ratio Test.

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Video Transcript

let's use the ratio test to see whether the Siri's is conversion or vibration. So let's go ahead and do knowthis by n then the ratios has wants you to look at the limit as n goes to infinity, and then you take the absolute value A and plus one over a end. So notice that the plus one is because the one is being added to that and not a M. So now let's go ahead and see if we could simplify this and plus one in the numerator That's just and plus one over five and plus one and then in the denominator years on end so and over five in Now let me bring this down over here so I don't need absolute value here because all the numbers and the fractions are all positive. So here we have n plus one over n. And then let me write this as I have five end and then I'LL rewrite this term here using my exploded properties I can rewrite This is five times five to the end and then I'LL cancel those five ends and that here. So I have a five on the denominator and then as I take the limiters and goes to infinity this fraction, if you want, you could use a little bit, tells Rule here. If I take the derivative of the N plus one, that's just one. If I take the derivative of end with respect to end issues one and then we still have this one over five, so we get an answer of one over five. This is less than one. So we conclude that the Siri's the one that we started with convergence by the ratio test, and that's our final answer.