MATH 201 Notes, Summer 2001

MATH 201 Things to know:  lim = 0  lim n1/n = 1  lim (1 + )n = ex  lim xn = 0 (|x| < 1)  lim x1/n = = 1 (x > 0)  lim = 0 (- < x < + ) ∞ ∞  The telescopic series: = 1  Geometric series: =  diverges, converges diverges  p-series: {  Alternating series:  Truncation error: let be convergent series with S = and Sn = , then trunc. error Tn = S - Sn  Alternating series estimation theorem ASET: If alternating series satisfies (a) & (b) of AST (Leibniz thrm.), then |Tn| < and sgn Tn = sgn(-1)n  A series is said to converge (a) absolutely if converges (b) conditionally if it converges but not absolutely, namely converges but diverges  Absolute convergence theorem: converges => converges  Rearrangement theorem for absolutely convergent series: Suppose (a) rearrangement of (b) converges absolutely Then converges absolutely and =  Power series centered at x=a: Converges for only one of the following cases: (a) the series converges only at center a (b) the series converges (absolutely) for every real number x R (c) There is a positive real number R such that: (i) the series converges (absolutely) for every (ii) the series diverges for every and every (iii) the series may converge or diverge at a ± R  Term-by-term differentiation: is differentiable and its derivative:  Term-by-term integration: is integrable and its integral: Summary of Tests:  n-th term test: - If lim an = 0 => converges - If lim an fails to exist or lim an 0 => diverges (test is necessary but not sufficient, ex: )  Integral test: converges (x N) ≥ converges  Direct comparison test DCT: suppose 0 ≤ ≤ (k 1), ≥ then (a) converges converges (b) diverges diverges  Limit comparison test LCT : suppose and where ≥ 0 & > 0 , then (a) lim = L > 0 => both series converge or both series diverge (b) lim = 0 & converges => converges (c) lim = ∞ & converges => converges (d) lim = ∞ & diverges => diverges  Ratio test: suppose and lim = L (possibly ) ∞ then (a) L < 1 => converges (b) L > 1 => diverges (c) L = 1 => is undecided (try other test)  Root test: suppose and lim = L (possibly ) ∞ then (a) L < 1 => converges (b) L > 1 => diverges (c) L = 1 => is undecided (try other test (n-term test))  Alternating series test AST (Leibniz thrm.): converges if: (all else => diverges) (a) ≥ for all k, and (b) lim = 0 (c) > 0