By Richard Beals

Compatible for a - or three-semester undergraduate direction, this textbook presents an advent to mathematical research. Beals (mathematics, Yale U.) starts with a dialogue of the houses of the true numbers and the idea of sequence and one-variable calculus. different themes comprise degree conception, Fourier research, and differential equations. it really is assumed that the reader already has an excellent operating wisdom of calculus. approximately 500 workouts (with tricks given on the finish of every) aid scholars to check their realizing and perform mathematical exposition.

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Let na or n · a denote a + a + · · · + a, where there are n summands. If n is a negative integer, we let n · a denote −(−n)a. ) We can assign to a rational number P1: KaD 0521840724c02 CY492/Beals 0 521 84072 4 June 18, 2004 14:21 Char Count= 0 2A. The Real Numbers 19 r = m/n the real number (m · 1)(n · 1)−1 , which we denote by r . With some labor one can prove the following, for any rationals r and s. (a) r = s if and only if r = s. (b) (r + s) = r˜ + s˜ and (r s) = r s. (c) r < s if and only if r < s.

Prove that the sequence {xn }∞ 1 converges and find its limit. 14. Let x1 = 1, x2 = 2, and xn+2 = 13 xn + 23 xn+1 , n = 1, 2, . . Prove that the sequence {xn }∞ 1 converges and find its limit. 15. Let positive a and x1 be given, and let xn+1 = (a + xn )/(1 + xn ), n = 1, 2, . . Prove that the sequence {xn }∞ 1 converges and find its limit. 16. Let positive a and x1 be given, and let xn+1 = 12 (xn + a/xn ), n = 1, 2, . . Prove that the sequence {xn }∞ 1 converges and find its limit. 3B. Upper and Lower Limits Suppose that {xn } is a bounded real sequence.

8: 2m test. Suppose that the real sequence {am } is nonincreasing and has limit 0: a1 ≥ a2 ≥ a3 ≥ . . ≥ an ≥ 0; Then ∞ 1 an converges if and only if the series ∞ 1 Example. Consider ∞ m=0 lim an = 0. n→∞ ∞ m=0 2m a2m converges. 1/n b , where b > 0.