Slide 11.3.pptx - Positive Term Series Integral Test...

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Positive Term Series Integral Test P-Series Basic Comparison Test
Positive Term Series A Positive Term Series is a series Σa n such that a n > 0 for every n. If no such M exists, the series diverges. If Σa n is a Positive-Term Series and if there exists a number M such that S n = a 1 + a 2 + … + a n < M for every n, then the series converges and has a sum S ≤ M.
The Integral Test Suppose f is a continuous, positive, decreasing function on [1,oo) and let a n = f(n). The series converges if and only if converges. In other words: a) If converges, then converges. b) If diverges, then converges. Roughly speaking, in order to prove whether a series converges/diverges, we need to show the corresponding integral converges/diverges.
The Integral Test In using the integral test, it is necessary to consider - If f(x) is not easy to integrate, use a different test for convergence or divergence - It is unnecessary to start the series or the integral at n = 1. We could start at n=k for every integer k > 0.
Illustration
Example 1 Does the series

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