Examples of General Elementary Series

Examples of General Elementary Series tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về tất cả các...

Examples of General Elementary Series Leif Mejlbro Download free books at Leif Mej lbro Exam ples of General Elem ent ary Series Calculus 3c- Download free eBooks at bookboon.com Exam ples of General Elem ent ary Series – Calculus 3c- © 2008 Leif Mej lbro & Vent us Publishing Aps I SBN 978- 87- 7681- 376- Download free eBooks at bookboon.com Calculus 3c-2 Contents Cont ent s Introduction Partial sums and telescopic series Simple convergence criteria for series 13 The integral criterion 41 Small theoretical examples 47 Conditional convergence and Leibniz’s criterion 49 Series of functions; uniform convergence 81 www.sylvania.com We not reinvent the wheel we reinvent light Fascinating lighting offers an ininite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges An environment in which your expertise is in high demand Enjoy the supportive working atmosphere within our global group and beneit from international career paths Implement sustainable ideas in close cooperation with other specialists and contribute to inluencing our future Come and join us in reinventing light every day Light is OSRAM Download free eBooks at bookboon.com Click on the ad to read more Calculus 3c-2 Introduction Introduction Here follows a collection of examples of general, elementary series The reader is also referred to Calculus 3b The main subject is Power series; but first we must consider series in general We shall in Calculus 3c-3 return to the power series Finally, even if I have tried to write as careful as possible, I doubt that all errors have been removed I hope that the reader will forgive me the unavoidable errors Leif Mejlbro 14th May 2008 Download free eBooks at bookboon.com Calculus 3c-2 Partial sums and telescopic series Partial sums and telescopic series Example 1.1 Prove that the series ∞ 1 − n+4 n+6 n=1 is convergent and find its sum We shall in this chapter only use the definition of the convergence as the limit of the partial sums of the series In this particular case we have N sN = n=1 = 1 − n+4 n+6 1 1 1 1 − − − − + + + ··· + N +2 N +4 1 1 − − + + N +3 N +5 N +4 N +6 The sum is finite, and we see that all except four terms disappear, so sN = 1 1 11 1 + − − → + −0−0= N +5 N +6 30 for N → ∞ It follows by the definition that the series is convergent and its sum is ∞ n=1 1 − n+4 n+6 = lim sN = N →∞ 11 30 Remark 1.1 Since N = n+4 n=1 N +4 n=5 n N and = n+6 n=1 N +6 n=7 n (finite sums with the same terms; check!), we get more well-arranged (the sum can be split, because it is finite) N sN = = = N 1 − = n + n + n=1 n=1 1 + + N +4 n=7 n N +4 n=5 N +4 − n=7 − n N +6 n=7 n 1 + + n N +5 N +6 1 1 1 11 + − − → + = 30 N +5 N +6 for N → ∞, etc Download free eBooks at bookboon.com Calculus 3c-2 Partial sums and telescopic series Example 1.2 Prove that the given series is convergent and find its sum ∞ (2n − 1)(2n + 1) n=1 Since we have a rational function in e.g x = 2n, we start by decomposing the term 1 1 = − (2n − 1)(2n + 1) 2n − 2n + Then calculate the N -th partial sum N sN = 1 = (2n − 1)(2n + 1) n=1 = N −1 n=0 1 − 2n + N 1 − 2n − n=1 N 2n + n=1 N 1 1 = − · 2n + 2 2N +1 n=1 Since the sequence of partial sums is convergent, sN = 1 1 − · → 2 2N + for N → ∞, the series is convergent and its sum is ∞ 1 = lim sN = (2n − 1)(2n + 1) N →∞ n=1 360° thinking Discover the truth at www.deloitte.ca/careers Download free eBooks at bookboon.com © Deloitte & Touche LLP and affiliated entities Click on the ad to read more Calculus 3c-2 Partial sums and telescopic series Example 1.3 Prove that the given series is convergent and find its sum, ∞ 2−1 n n=2 We get by a decomposition, n2 1 1 1 = = · − · −1 (n − 1)(n + 1) n−1 n+1 Se sequence of partial sums is then N sN N N = 1 1 = − n −1 n=2 n − n=2 n + n=2 = = N −1 1 − n N +1 1 + + N −1 n=1 n=3 n=3 n n (the same insides, check the first and the last terms) − N −1 n=3 1 + + n N N +1 (remove some terms) 1 1 − · − · (cancel the two identical sums) N N +1 for N → ∞ → It follows by the definition that the series is convergent and its sum is = ∞ = lim sN = 2−1 N →∞ n n=2 Example 1.4 Prove that the given series is convergent and find its sum √ ∞ √ n+1− n √ n2 + n n=1 This is a nontypical case, though one may still copy the method of decomposition since √ √ n2 + n = n + · n, it follows by a division that √ √ √ √ 1 n+1− n n+1− n √ = √ √ =√ −√ n n+1· n n+1 n +n Then calculate the sequence of partial sums, √ N N N √ n+1− n 1 √ √ − √ sN = = 2+n n n +1 n n=1 n=1 n=1 N = √ − n n=1 N +1 n=2 1 1 √ =√ −√ =1− √ n N +1 N +1 Download free eBooks at bookboon.com Calculus 3c-2 Partial sums and telescopic series Since the sequence of partial sums is convergent sN = − √ →1 N +1 for N → ∞, the series is also convergent and its sum is √ ∞ √ n+1− n √ = lim sN = N →∞ n2 + n n=1 Remark 1.2 We see from the expression of the sequence of partial sums that the convergence is very slow, so it is not a good idea here to use a pocket calculator Example 1.5 Prove that the given series is convergent and find its sum, ∞ n=1 2n + + 1)2 n2 (n We first decompose, (n2 + 2n + 1) − n2 1 2n + = = 2− + 1) n2 (n + 1)2 n (n + 1)2 n2 (n This gives us the sequence of partial sums N sN = N N 1+ = 1− n2 n=2 N +1 n=2 n2 N N = N 2n + 1 1 = − = − (n + 1)2 2 n n (n + 1) n n=1 n=1 n=1 n=1 − 1 + n (N + 1)2 n=2 →1 (N + 1)2 for N → ∞ It follows by the definition that the series is convergent and its sum is ∞ n=1 2n + = lim sN = N →∞ + 1)2 n2 (n Example 1.6 Prove that the given series is convergent and find its sum, ∞ 3n + n(n + 1)(n + 2) n=1 We get by a decomposition that 1 3n + = − − n(n + 1)(n + 2) n n+1 n+2 Download free eBooks at bookboon.com ... follows a collection of examples of general, elementary series The reader is also referred to Calculus 3b The main subject is Power series; but first we must consider series in general We shall in...Leif Mej lbro Exam ples of General Elem ent ary Series Calculus 3c- Download free eBooks at bookboon.com Exam ples of General Elem ent ary Series – Calculus 3c- © 2008 Leif Mej... and telescopic series Simple convergence criteria for series 13 The integral criterion 41 Small theoretical examples 47 Conditional convergence and Leibniz’s criterion 49 Series of functions;