Problem 3 Easy Difficulty

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} ( - 1)^n nx^n $

Answer

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Video Transcript

Okay, So our radius of convergence, remember, Will you limit as n goes to infinity Absolute value of an over and plus one. These guys are R A in terms So this is going to be limit as n goes to infinity. Ah, see, just in over in plus one that's one. So our radius of convergence is just one for the interval of convergence. We want to see what happens when we plug in one into R sum and we want to figure out what happens when you plug in minus one into R sum So access minus one, then our terms they are just going to be minus one times minus one times in. So what is going to be something from in equals? One to infinity of n. So this is definitely going to diverge. And similarly, when X is equal to one, we're just going to have some from n equals one to infinity of minus one to the end times in. And that's still going to diverge because their terms are not even going to zero. So are in points are goingto be thrown out. Those aren't any good. So our interval of convergence we have to leave open like this interval of convergences interval from minus one toe, one open