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Numerade Educator

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Problem 42 Medium Difficulty

Suppose that the radius of convergence of the power series $ \sum c_n x^n $ is $ R. $ What is the radius of convergence of the power series $ \sum c_n x^{2n} ? $

Answer

Radius of convergent is $\sqrt{R}$

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Video Transcript

we can use the ratio tests on this sum here. So lim his n goes to infinity absolute value of and plus one over. And we want this to be less than one by and we mean this whole chunk here, including the X values. This is Linda as n goes to infinity. Absolute value of X to the two times in class one times C n plus one over X to the two n times cnn X to the two and plus one. If we expand that, that's X to the two n plus two. So the X to the two and we'LL cancel out with this X to the two and and we'LL just leave us with X squared And over here we still have the c n plus one overseeing and again we want this to be less than one. We can write this as absolute value of X squared is less than limit as n goes to infinity of absolute value of C n o ver si n plus one. So if we just flip this, multiply both sides by the reciprocal of this limit and since that's a positive number, that's not going to affect this inequality sign here. This limit here should be the radius of convergence. For this sum, as we saw from the previous exercise, That's our Okay, So absolute value of X squared is the same thing as absolute value of X squared. So the absolute value of X squared is less than are Can we get that absolute value of X is less than a square root of are And this tells us that the square root of are is that radius of convergence for this power Siri's