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Problem 24 Easy Difficulty

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

$ \sum_{n = 1}^{\infty} \frac {n^2x^n}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot \cdot \cdot (2n)} $

Answer

$R=\infty$
Interval of convergence: $(-\infty, \infty)$

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Video Transcript

okay to figure out the radius of convergence, you might, you know, just remember, this shortcut here at the radius of convergence is limit as n goes to infinity of absolute value of a N over a a and plus one Where the And is this chunk here, not including the X values. If you don't remember that shortcut, then you could just use the ratio test on this whole chunk here and figure out when you're less than one. Okay, but this is just a little shortcut. So this is limit as n goes to infinity of absolute value of and squared over two times, four times that that that times to end. So that's just our way in terms. And we're dividing by n plus one so multiplying by they're reciprocal. So two times for attempt at that that times to end times, times two in class one and we're dividing by and plus one squared. Okay, so this is Lim. His n goes to infinity in squared, divided by in plus one squared. That's the same thing as in Divided by n Plus One squared. And now this too is going to cancel it. That, too, that for is going to cancel with that, for we get all this whole chunk to cancel out with this whole chunk. So now we just have two times in plus one here. Okay? As n goes to infinity and over in plus one is one One squared is one This is going to be infinity. So we have one times infinity. So the radius of convergence is infinity. Therefore, the interval of convergence is the interval from minus infinity to infinity.

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