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MATH231 Lecture Notes 4

# MATH231 Lecture Notes 4 - 1 Convergence of Series Last time...

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1 Convergence of Series Last time we introduced the first important test: the integral test. The integral test says the following: Theorem: Integral Test Suppose f ( x ) is continuous function which is eventually decreasing and positive, and a n = f ( n ). Then either 1 a n 1 f ( x ) dx both converge, or they both diverge. This is a convenient test because it ALWAYS gives an definite outcome (assuming that one can test convergence of ONE of the series. Let’s look at some examples: Example: The series n =1 1 n p Converges if p > 1 and diverges if p 1. This is easy to see by applying the integral test. It is easy to see that f ( x ) = 1 x p is decreasing for p > 0, and is continuous on [1 , ). Computing the integral lim R →∞ R 1 dx x p = lim R →∞ x 1 - p 1 - p | R 1 (1) = lim r →∞ 1 1 - p (1 - R 1 - p ) = 0 p > 1 p < 1 (2) In the case p = 1 one gets lim R →∞ R 1 dx x = lim R →∞ ln( x ) | R 1 = lim R →∞ ln( R ) = Aside: The function f ( p ) = n =1 1 n p is called the Riemann Zeta function. There is a conjecture (the Riemann hy- pothesis) that says that f

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MATH231 Lecture Notes 4 - 1 Convergence of Series Last time...

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