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# notes3 - Monotone Sequences Maybe you remember hearing...

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Monotone Sequences Maybe you remember hearing about how the real numbers can be broken up into two types of numbers. There are algebraic numbers such as 1 , 3 , 2 , 1 / 2 , p 2 + 3 i.e., numbers from some sort of algebraic manipulation and then there are transcendental numbers. I really only know two them, π and e (This is not the same as the difference between rational and irrational numbers. All rational numbers are algebraic and so are some irrational). If you threw a dart at the real line, you would have a 100 percent chance of hitting a transcendental number - yet I only know two. The point is, we just don’t know what to call most of the numbers on the real line. We could use more greek letters besides π , but there just isn’t enough. So we won’t call them anything. We just will remember that they exist and, in fact, are what makes up most of the real line. Math 9C Summer 2011 (UCR) Pro-Notes June 21, 2011 1 / 15

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So, if we’re given a random sequence and we know it converges, most likely we just won’t have a name for what it converges to. Remember this: All we can hope to know about an arbitrary infinite sequence is determining if it converges or not. If we get lucky, maybe we’ll be able to find out what it converges to, but most likely that’s just not possible. So our job at first is learning all the tricks we can to determine when a series converges or when a series diverges. Math 9C Summer 2011 (UCR) Pro-Notes June 21, 2011 2 / 15
Monotone Convergence theorem The main idea of this theorem says: I If all the terms of a sequence { a k } are moving to the right on the number line, but are never larger than some number M , then the sequence must converge.

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notes3 - Monotone Sequences Maybe you remember hearing...

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