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Problem 39 Hard Difficulty

Show that if $ \lim_{n \to \infty} \sqrt[n]{\mid c_n \mid} = c, $ where $ c \not= 0, $ then the radius of convergence of the power series $ \sum c_n x^n $ is $ R = 1/c. $

Answer

$$R=1 / c$$

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Video Transcript

this problem is basically just applying the root test. So this is the sum that we're looking at here is going to do in the root test. We take the in through of this chunk here rather the in threat of the absolute value of all that stuff. So the in through of the absolute value of X to the end, it's just going to be the absolute value of X. I can and the stuff we have leftover is thean through of absolute value of CNN. But we're given that the as in goes to infinity the in through it of the absolute value of C N is equal to C. Okay, so now we have absolute value of X time, see, and similar to the ratio test, we want this to be less than one and since he is non zero, we know that we're allowed to divide by sea. Now we have absolute value of X is less than one oversee and since he is positive since the end through it of ah, positive number is always going to be positive, we know that dividing by sea is not gonna affect this inequality sign. If we're dividing and multiplying by a negative number than this inequality sign would have to be flipped since he is positive. This is less then and this is still less than Okay, So this tells us that we get convergence when absolute value of X is less than one Oversee. So exes trapped between minus one, oversee and see. So we get convergence. A cz long his ex is somewhere between here so we can see that the length of our interval of convergence is one. Oversee minus minus one. Oversee, so to oversee. So the radius of convergence is half of that. So radius of convergence is one oversee, which you could probably just see from looking at this and then the only other place we could possibly get Convergence is if we have equal toe one here so similar to the ratio test. If it's equal the ones and the test is not conclusive, you could get convergence and you might not get convergence. But whether or not we include the end points minus one oversee and one oversee for interval of convergence, it's not going to affect the length of that interval. So regardless of whether or not we include the end points, the radius of convergence is going to stay at one oversee