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Numerade Educator

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Problem 33 Easy Difficulty

Is it possible to find a power series whose interval of convergence is $ [0, \infty)? $ Explain.?

Answer

$(-\infty, \infty)$

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Video Transcript

you should always be ableto right. A power Siri's in this form. Okay. Where the sum could go from zero to infinity or wondering Cindy er, whatever. Okay. And then, if you have your power, Siri's in this form, then the interval of convergence is going to look something like this so we could get Ah, we might include the end points. We might not, but at least on this open interval, we will get convergence. So we will get convergence here and then the end points. It depends on the power. Siri's okay and are here. Our is the radius of convergence for the power Siri's. So the radius of convergence is one half of the length of the interval of convergence. So here the length of the interval of convergence is infinity. That means that the radius of convergence in this case is infinity. So since sea is just some finite number C minus, infinity is minus infinity and C plus, infinity is infinity. So in other words, if we got convergence here, then we would need to get convergence here