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Problem 8 Easy Difficulty

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

$ \sum_{n = 1}^{\infty} n^nx^n $

Answer

interval of convergence $[0,0],$ radius $R=0$

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Video Transcript

okay for this problem. Our radius of convergence are is lim a as n goes to infinity into the end over in plus one to the end plus one. Okay, so we can rewrite that as limited n goes to infinity of in over in plus one to the power of end and then times one over in plus one. Okay, so in divided by n plus one to the power of n that's one over e and then we're left with limit as in approaches infinity of one over in plus one and this is zero. So we have won over E which is just some finite number time zero So we end up with zero. So the radius of convergence here is zero. The interval of convergence. We just checked the end points on this case. We're just checking zero here and we're checking to see whether or not this is going to converge. There's just adding up zero a bunch of times. So this this is going to be zero to that is going to work. Case will include zero, but nothing else will be included in our interval of convergence Are interval of convergence is just zero all by itself

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