By Richard Beals

Appropriate for a - or three-semester undergraduate direction, this textbook presents an creation to mathematical research. Beals (mathematics, Yale U.) starts with a dialogue of the homes of the genuine numbers and the speculation of sequence and one-variable calculus. different issues comprise degree idea, Fourier research, and differential equations. it's assumed that the reader already has a great operating wisdom of calculus. approximately 500 routines (with tricks given on the finish of every) aid scholars to check their realizing and perform mathematical exposition.

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Prove (3) to (6), using Exercises 10–12 of Section 2A. 7. Suppose that {z n }∞ 1 is a complex sequence with limit z 0 . Prove directly from the definitions that (a) limn→∞ z n2 = z 02 and (b) limn→∞ z nk = z 0k , k ∈ IN. 8. Suppose that |z| < 1. Prove √ that limn→∞ z n = 0 9. )1/n . 10. 2 is equivalent to the Least Upper Bound Property. 2 is valid. Show that O6 is a consequence of these assumptions. P1: IwX 0521840724c03 CY492/Beals 0 521 84072 4 June 18, 2004 14:35 Char Count= 0 3B. Upper and Lower Limits 33 11.

Suppose that {z n }∞ 1 is a complex sequence with limit z and suppose that {an }1 is a positive sequence such that limn→∞ (a1 + a2 + . . + an ) = +∞. Prove that a1 z 1 + a2 z 2 + . . an z n = z. n→∞ a1 + a2 + . . + an lim 6. Prove (a) limn→∞ x n /n k = +∞ if x > 1 and k ∈ IN; (b) limn→∞ n k x n = 0 if |x| < 1 and k ∈ IN; (c) limn→∞ x 1/n = 1 if x > 0. 7. 718 . . : n! n 100 n n/2 2n n 2 n nn . P1: IwX 0521840724c03 CY492/Beals 0 521 84072 4 June 18, 2004 42 14:35 Char Count= 0 Real and Complex Sequences 8.

Thus |an | ≤ r n for n ≥ N , and the Comparison Test implies convergence. 3, so the terms do not have limit 0. ∞ 1/n Example. = 1, so the Root 1 1/n. By Exercise 5 of Section 3F, limn→∞ n ∞ 2 Test is inconclusive. The same is true for the series 1 1/n . P1: IwX 0521840724c04 CY492/Beals 0 521 84072 4 June 18, 2004 14:45 50 Char Count= 0 Series ∞ 2 Neither the Ratio Test nor the Root Test tells us whether ∞ 1 1/n or 1 1/n converges or diverges. The terms of these series are positive, so all we need determine is whether or not the partial sums are bounded.