MATH 401 INTRODUCTION TO ANALYSIS-I, SPRING TERM 2016, PROBLEMS 12 Return by Monday 11th April Generally given a sequence xn we use n xm to denote x1 + x2 + + xn . m=1 Throughout dene an = (1 + 1/n)n (n N), bn = (1 1/n)n (n = 2, 3, 4, . . . ) and n 1 cn = 1 + m! m=1 1. (i) Prove that for m N, 2m1 m!. (ii) Prove that 2 cn 1 + n m=1 (n N). 1 2m1 < 3. (iii) Prove that cn is increasing, and hence converges. Let e = limn cn . Prove that 2 e 3. 2. The binomial theorem is useful in this question. (i) Prove that for n N, )( ) ( ) n 1 ( 1 2 m1 an = 1 + 1 1 ... 1 . m! n n n m=1 (ii) Prove that an is increasing. (iii) Prove that an e. (iv) Prove that an converges. Let e = limn an . 3. (i) Prove that 1 < bn /an n/(n 1) (n = 2, 3, . . . ). (ii) Show that bn also converges to e . 4. (i) Suppose that 1 r n (r, n N). Prove that )( ) ( ) r 1 ( 1 2 m1 1+ 1 1 ... 1 an e. m! n n n m=1 (ii) Prove that (iii) Prove that e = e. r 1 1+ e e. m! m=1