Problem 1 Medium Difficulty

If the radius of convergence of the power series $ \sum_{n = 0}^{\infty} c_n x^n $ is 10, what is the radius of convergence of the series $ \sum_{n = 1}^{\infty} nc_n x^{n -1}? $ Why?

Answer

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Video Transcript

to solve this problem, they need to consider the use Racial test toe performs the ritual test they need. Ah, Siri's a m So they just who's a and e kowtow the series that we even defends the radios of convergence and they can wreck down Ah lim on goto infinity and abstruse while you and Class A and class went away and and they use AA number this limit member to show it's radios of Karajan's on DA This is e kowtow. They made, uh, on go to infinity Nu Merita part is a plus one. So we have on press one times c and press one on DH terms angst to the power and managed to grasp is accepted. Oh, uh a n a a c n i times AKs and to the power and menace one absolute value. So they want to simplify this equation. What we kite is limit on go to infinity and pass away Oh, and on the temps see and place well oh, I see times likes to the power and divide by X to the power and minds when it was left is X So we want this power test. Toby, it's this racial tested Toby smaller, smaller than one. So we got a creation. You are to be highways led a go to infinity. This number is very, very close toe so we can ignite and from the condition. So radios of convergence of the palace era power Siri's of this serious is turn. So what we cat is say and plastic man times X o or CNN is ICO toe the force of a signal the first part because they will goto eight veil goes to one And what a left of this party is? Oh, attend Time's likes from this condition and because of this number is smaller than land. So what a high is axe Absolute value is equal to ten eyes smaller than ten Sorry. So the radio's of convergence of these a n Siri's is also ten. The idea