Mathematics 1

Advanced calculus with linear analysis - download pdf or read online

By Joseph R. Lee

ISBN-10: 0124407501

ISBN-13: 9780124407503

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7. From Problems 5 and 6, prove lim-—7== = e. nWw! In Problems 8-34, determine convergence or divergence of the given series. If an alternating series converges, test for absolute or conditional convergence. 4 Additional Convergence Tests—Alternating Σ wιν -. « (-2)» 12. Σ ^ - 20. Σ 21. Σ 22. Series (-1)ηη ηΛ — TT¿ (-1)Λ1η(η) ^ , 1η(η + 1) - 1η(η - 1) η-2 is. Σ - · 14. 23. 1 Ση\η{η) » •A Inn οο 24. η=2 η=1 25. 16. Σ Λ/Π2 +1 26. η=1 17. οο Σ η=1 27. 52 α„, where 28. 3(η+1)/ -1 +1 ——- 19. n+ η 00 18.

Lim I ( 1 + \ + I + - - - + - ) - In n 1 = lim Σ ck. n->ooL\ 2 3 nj J n^ k=1 This number is called Euler's constant or Mascheroni's constant. 577216. It is not known if it is transcendental or even if it is irrational. 8. 10) gives a convenient way of estimating the difference between the partial sums and the sum of an infinite series. Prove the following: If the series ΣΓ=ι ak converges by the integral test [where f(k) = au], and if A = An + Rn, where A is the sum of the series and An the nth partial sum, then | An | < Γ fix) dx.

Obviously an infinite number of points in the sequence cluster about 0, but 18 II. Sequence Spaces and Infinite Series that is neither the limit, the limit superior, nor the limit inferior. 6 DEFINITION The point a is a limit point, or cluster point, of the sequence {an\ if every interval (a — e, a + e) contains points of {an} other than a itself. (We leave for an open argument the question of whether the sequence 0, 1, 0, 1, 0, 1, . . ) PROBLEMS 1. 5 for limit superior. In Problems 2-13, an is given.

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Advanced calculus with linear analysis by Joseph R. Lee

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