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Problem 9 Medium Difficulty

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

$ \sum_{n = 1}^{\infty} \frac {x^n}{n^44^n} $




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Video Transcript

Hey, to figure out the radius of convergence, we do limit as n goes to infinity of absolute value of a N over absolute value of and plus ones, absolutely of this whole thing where the ends air just one over into the fourth times for the end. This is limit as n goes to infinity Ah, in plus one to the fourth times four to the end, plus one divided by into the fourth times for it to the end. So this's limit as n goes to infinity of thin plus one over in to the fourth and then for the n plus one divided by four to the end Oops, sorry. I forget that for the M plus one divided by four the end that's going to cancel out just before and then and plus one over in as n goes to infinity and plus one over n goes toe wants This is one to the fourth, which is just one. So one times for so just get for here. So four is the radius of convergence Here for the interval of convergence, we need to check whether or not four works when we plug it in here and whether or not minus four works when we plug it in here. So when axes equal to minus four, then we have minus four to the end over into the fourth times for the end. Sorry for the bad hand writing. Okay, so minus four to the end, divided by four to the end, that's going to turn into a minus one to the end because minus forty and is minus one of the end times for the end and then four to the and will cancel out before the end that we have down there. Okay? And then this will converge, but the alternating signed test. And if we look at, if we look to see what happens when X is equal to four, then we would end up with for the end divided by four to the end. So those would cancel. And we just have one over and to the fourth. Since four is a number that is larger than one, this would be something convergent. So this would also converge. Okay, that works. So minus one works are starting minus four works. When you plug it in for works, when we plug it in. So We include both minus four and foreign R Interval of convergence. So we use these close brackets. Tau show that we are including them and that's gonna be our answer.

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