Intro to Analysis in-class_Part_49

Intro to Analysis in-class_Part_49 - 7.2 Numerical Series...

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7.2 Numerical Series and Sequences 103 Shifting one unit to the right, the region under the graph is contained in the rectangle, so 0 A n f ( n ) , for n N. This gives N + n X k = N +1 f ( k ) Z N + n N +1 f ( x ) dx N + n - 1 X k = N f ( k ) Z N + n - 1 N f ( x ) dx and so the two sequences converge or diverge together. Theorem 7.2.18 (p-series) . 1 n p converges iﬀ p > 1 . Proof. For p 0, 1 n p is decreasing, so apply the integral test to Z 1 1 x p dx = lim r →∞ Z r 1 1 x p dx = lim r →∞ r 1 - p - 1 1 - p , p 6 = 1 , lim r →∞ log r, p = 1 . For p = 1, log r → ∞ and both diverge. For p > 1, r 1 - p 0, so both are ﬁnite. For 0 p < 1, r 1 - p → ∞ , so both diverge. Finally, consider p < 0 and put q = - p > 0. Then n q diverges by n th term test. Theorem 7.2.19 (Asymptotic comparison test) . If lim | a n | | b n | = L , where L (0 , ) then X | a n | converges ⇐⇒ X | b n | converges . Proof.

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Intro to Analysis in-class_Part_49 - 7.2 Numerical Series...

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