Math 156: Exam 3 Review - Sequences and Series, Study notes of Calculus

Download Math 156: Exam 3 Review - Sequences and Series and more Study notes Calculus in PDF only on Docsity! Math 156 Review topics for exam 3 Sequences: sequence notation: ann=1 ∞ , an = fn , n = 1, 2, . . . or simply a1, a2, . . .  developing a formula for a given sequence - even numbers, odd numbers, exponentials, powers limits of sequences: techniques such as factoring out the largest term in a sum, L’Hopital’s rule combining sequences: limit laws Comparison of basic function classes: who goes faster to infinity. Series: Notation: ∑ n=1 ∞ an , a1 + a2 + a3 +. . . The sequence snn=0 ∞ of partial sums: sn =∑ i=0 n ai The sum of an infinite series, what it means: lim n→∞ sn = S and we write ∑ i=0 ∞ ai = S Convergence/divergence, absolute convergence Any increasing sequence either has limit infinity or converges. Hence: A bounded increasing sequence must converge. Similarly for decreasing sequence, e.g. any decreasing sequence of positive terms must converge. Special series: Geometric series, finding the sum of a geometric series p − series, ∑ n=0 ∞ 1 np Analysis of series for convergence: Convergence is independent of any number of terms at the beginning of the series. So any comparison that is "eventually" true (i.e. is true for n large enough) can be used. If series ∑ n=0 ∞ an converges, must have an → 0 Basic comparison for positive series: Combining and breaking up convergent series Detailed tests: Integral test: an = fn , fx positive and decreasing ∑ n=1 ∞ an converges if and only if ∫ 1 ∞ fxdx converges 1