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Section14and15notes - Lectures 14 and 15 Infinite series convergence tests and power series Definitions sequence of partial sums infinite sum

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Unformatted text preview: 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 ±∞. √ √ n + 1 − n. Examples: geometric series, ∞ n(n1+1) , n=1 Definition: Cauchy criterion for infinite sums. Proposition: If an converges then an → 0. Counterexample to show the converse of the previous proposition does not hold (harmonic series). Comparison test: Let an be a series with an ≥ 0 ∀n. If an converges and |bn | ≤ an for all n, then bn converges. If an diverges to ∞, and bn ≤ an for all n, then bn diverges to ∞. 1 . Example: 2n +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 an > 0. 1. If lim sup 2. If lim inf an+1 an an+1 an < 1, then the series converges absolutely. > 1, then the series diverges. 3. Otherwise, the test gives no information (i.e. there are series with both behaviors). an , let α = lim sup |an |1/n . Root test: Given the series 1. If α < 1, then the series converges absolutely. 2. If α > 1, then the series diverges. 3. Otherwise α = 1 and the test gives no information. n n! 1 1 , , , . Examples: 2n 2n n2 n Integral test: Let f be a continuous function defined on [0, ∞) that is both positive and decreasing. Then f (n) converges iff n lim n→∞ exists as a real number. Application: the p-series 1 np f (x)dx 1 converges iff p > 1. 1 Alternating series test: If (an ) is decreasing and an > 0 for all n, and if lim an = 0, then (−1)n an converges. cos(nπ ) √ Examples: alternating harmonic series, . n Definition: power series, radius of convergence. Ratio criterion: The radius of convergence R of a power series an xn is equal to an lim an+1 , provided the limit exists. Examples: xn , 1n x, n 1n x. n! 2 ...
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This note was uploaded on 10/16/2011 for the course MATH 34 taught by Professor Wiley during the Winter '11 term at UC Merced.

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Section14and15notes - Lectures 14 and 15 Infinite series convergence tests and power series Definitions sequence of partial sums infinite sum

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