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# notes10 - The ratio test The next test we'll learn about is...

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The ratio test The next test we’ll learn about is called the ratio test. It’s really the best test out there. And we’ll see later that it’s also a very important test when we start to study power series. The idea behind the ratio test is this: In a geometric series the ratio of two consecutive terms was always r , by this I mean if a k = r k , then a k +1 a k = r k +1 r k = r for every k . And we know that a geometric series converges if | r | < 1 and diverges if | r | ≥ 1. The ratio test says: if you don’t have a geometric series, but you do have that “in the limit” the ratio of two consecutive terms is some number ρ (funny Greek letter for r called rho), our series will behave similar to a geometric series. Math 9C Summer 2011 (UCR) Pro-Notes October 14, 2011 1 / 1

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Theorem (The ratio test) Let k =1 a k be an inﬁnite series. Set ρ = lim k →∞ ± ± ± ± a k +1 a k ± ± ± ± Then I If ρ < 1 , the series converges (it even converges absolutely!). I
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notes10 - The ratio test The next test we'll learn about is...

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