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Intro to Analysis in-class_Part_47

# Intro to Analysis in-class_Part_47 - 7.2 Numerical Series...

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7.2 Numerical Series and Sequences 99 NOTE: series are not multiplicative like sequences are: X a n b n 6 = X a n X b n , because 1 + a 1 b 2 + a 2 b 2 + . . . a n b n 6 = (1 + a 1 + · · · + a n )(1 + b 2 + · · · + b n ) . (More cross terms on right.) Theorem 7.2.7 (Increasing & bounded) . If 0 a n , n , then a n converges iff the partial sums are bounded. Proof. ( ) lim s n exists = ⇒ { s n } bounded. ( ) s n = s n - 1 + a n s n - 1 , so monotone. Then { s n } bounded implies { s n } convergent by completeness. Definition 7.2.8. a n is absolutely convergent iff | a n | converges. a n is conditionally convergent iff | a n | diverges but a n converges. Example 7.2.3. 1. For a positive-term series, convergence absolute convergence. 2. ( - 1) n 2 n and ( - 1) n n ! are absolutely convergent, since 1 2 n and 1 n ! are conver- gent. 3. ( - 1) n n is conditionally convergent, since the harmonic series diverges. Most comparison tests actually establish absolute convergence. In a moment, we’ll see that absolute convergence implies convergence, so this is a stronger result.

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