# Joseph R. Lee's Advanced calculus with linear analysis PDF By Joseph R. Lee

ISBN-10: 0124407501

ISBN-13: 9780124407503

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2 8. Sequences Spaces an = (-1)»M. - ; J + 19 C0S n = 1,2,3,.... Y> 9. an= (-l)«(l ~ J + ( ~ 1 ) n + 1 ( 1 + ~)> n = 1,2,3,.... 10. η7Γ « . η7Γ . ^ a n = 3 sin —■ + 2 cos —, 11. a n = 2(-Dn + ( - 1 ) » Í 2 + J j , n = 1, 2, 3, . . 12. an = 2(-Dn + S*" 1 ^ 1 ( l + - ) , n = 1, 2, 3, . . 13. 14. 15. „ Λ« n = 1, 2, 3, . . η7Γ 1 / . ητ , \ „~ « an = - l sin — + cos — I , n = 1, 2, 3, . . n \ 2 2 / Prove that if lim supn-*oo «n = lim infn^oo «n, then limn-oo «n exists and equals the common value of the other two.

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. w=1 (-1)"η! ύ if n is odd, if n is even. ·(2η-1) 33 -1)" Σ ( 1+(1/η) «_1« ">0 " Σ 29.

2 DEFINITION The sequence {an} is said to converge to a finite number a if lim^^ an = a; that is, if given e > 0, there exists an N such that | an — a\ < e if n > N. If no such a exists, the sequence diverges. If for every M > 0 there is an N such that \ an\ > M when n > N, we write lim«^ an = <». If it is necessary to distinguish between + oo and — <χ>} we require an > M in the first case, and an < — M in the second, when n > N. A little reflection indicates that there are in general two ways in which a sequence of real numbers may diverge; we may have linin^oo an = oo, or the numbers in {an\ may oscillate in some way to prevent \ an — a \ remaining small for any a.