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section14and15notes

section14and15notes - Lectures 14 and 15 Innite series...

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Lectures 14 and 15: Infinite series, convergence tests, and power series Definitions: sequence of partial sums, infinite sum. Infinite sums which converge, diverge, or diverge to ±∞ . Examples: geometric series, n =1 1 n ( n +1) , n + 1 - n . Definition: Cauchy criterion for infinite sums. Proposition: If a n converges then a n 0. Counterexample to show the converse of the previous proposition does not hold (harmonic series). Comparison test: Let a n be a series with a n 0 n . If a n converges and | b n | ≤ a n for all n , then b n converges. If a n diverges to , and b n a n for all n , then b n diverges to . Example: 1 2 n +1 . Definition: absolute convergence. Proposition: Absolutely convergent series are convergent. Counterexample to show the converse of the previous proposition does not hold (alternating harmonic series). Ratio test: Suppose each a n > 0. 1. If lim sup a n +1 a n < 1, then the series converges absolutely. 2. If lim inf a n +1 a n > 1, then the series diverges. 3. Otherwise, the test gives no information (i.e. there are series with both behav- iors). Root test: Given the series
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