MAE101 CAL v2 chapter 7 sequences and series

ĐẠI HỌC FPT CẦN THƠ Chapter Sequences and Series (Page 427-509, Calculus Volume 2) Dr Tran Quoc Duy Email: duytq4@fpt.edu.vn Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Topics: • 5.1 Sequences • 5.2 Infinite Series • 5.3 The Divergence and Integral Tests •5.4 Comparision Tests •5.5 Alternating Series •5.6 Ratio and Root Tests Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Definition A sequence is a list of numbers written in definite order: a1, a2,…,an ,… a1 is first term, a2 is second term,…, an is the nth term Notation: The sequences {a1, a2,…,an,…} is also denoted by {an} Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example Find the first terms of Fibonacci sequence {fn} f1 = f = 1, f n = f n −1 + f n − 1, 1, 2, 3, 5, 8, … Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example Find a formula for the general term an of the sequence: {3/4, -4/8, 5/16, -6/32, 7/64,…} Solution: an = (-1)n-1(n+2)/2n+1 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Definition The sequence {an} has the limit L, written limn→∞an=L, if we can make the terms an as close to L as we like by taking n sufficiently large If limit L exists and is finite, we say {an} converges; otherwise, it diverges Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Theorem lim can = c lim an n → n → lim (an  bn ) = lim (an )  lim (bn ) n → n → n → lim (an bn ) = lim (an ) lim (bn ) n → n → n → lim (an / bn ) = lim (an ) / lim (bn ) provided lim bn  n → ( n → lim (f (an )) = f lim (an ) n → n → ) n → n → provided that f is continuous Theorem Suppose that f :R→R such that limx→∞f(x) = L Then the sequence {an = f(n)} also converges to L Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example Investigate the convergence of the sequences: a) an = (-1)n b) an = (ln n)/n c) an = n!/nn Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Theorem If limx→∞ |an|=0 then limx→∞ an=0 Example Investigate the convergence of the sequences: (a) (−1) n n (b) sin n n Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Theorem 0 if | r |  n lim r = 1 if r = n → diverges if | r |  Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ 5.4 Comparison test Example  Is the series  n convergent? n =0 − n  Compare the given series with 3n c = lim n n → − n 1  n n =0 =1 By Integral Test   1 1, then the series is divergent •If L = 1, the Ratio Test is inconclusive; that is, no conclusion can be drawn We have to use other tests (e.g., integral, comparison, alternating series Tests) Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ 5.6 Ratio and Root test Example ( −1)n n! Is  convergent? n n =1  an +1 ( −1)n +1(n + 1)! 5n L = lim = lim n +1 n n → a n → ( − ) n! n ( −1)(n + 1) = n → = lim L > → the given series is divergent Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ 5.6 Ratio and Root test Example Investigate the convergence of the series: (−1) n n (a)  n n =1   nn (b)  n =1 n! Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ 5.6 Ratio and Root test The Root Test (Cauchy)  Given a series  a Let n n =k n L = lim an n → Then, • If L < then the given series converges absolutely (hence, converges) • If L > 1, then the given series diverges •If L = 1, the Root Test is inconclusive; that is, no conclusion can be drawn We have to use other tests (e.g.,the integral, comparison, alternating series Tests) Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example  nn Is  1+ n convergent? n =1 L = lim an n → 1/ n n n = lim 1+ 2n n → = lim n → n 31/ n + 1/ n  = = L > → the given series diverges Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Guidelines for convergent/divergent testing •If limn→∞an≠0 the series ∑ an diverges •If an is of the form 1/np or xn, use the property of these special series • If an is a rational fraction or roots of polynomials, use the comparison/limit comparison Test •If in the expression of an has ! or xn , use the Ratio Test •If an = (bn)n, use the Root Test or Ratio Test •If an is alternating, use the alternating series test •If an = f(n) is positive and decreasing, use the Integral Test Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ... HỌC FPT CẦN THƠ Topics: • 5.1 Sequences • 5.2 Infinite Series • 5.3 The Divergence and Integral Tests •5.4 Comparision Tests •5.5 Alternating Series •5.6 Ratio and Root Tests Dr Tran Quoc Duy... a2 + + an + , is called a series The sum of first n terms: n Sn =  ak = a1 + a2 + + an k =1 is called the nth partial sum If the sequence {Sn} is convergent, then series is called convergent... Otherwise, the series is called divergent Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ 5.2 Infinite Series Example Find the nth partial sum of the following series and determine