1. Let (a n ) and (b n ) be monotone sequences. Prove or give a counterexample. (a) The sequence (c n ) given by c n = ka n is monotone for any k ∈ . (b) The sequence (c n ) given by c n = a n /b n is...

1. Let (a_n) and (b_n) be monotone sequences. Prove or give a counterexample.

(a) The sequence (c_n) given by c_n
= ka_n
is monotone for any k ∈
.

(b) The sequence (c_n) given by c_n
= a_n
/b_n
is monotone, where b_n
≠ 0 for all n ∈
.

2. Let (s_n) be the sequence defined by s_n
= (1 + 1/n)ⁿ. Use the binomial theorem to show that (s_n) is an increasing sequence with s_n
<>_n) is convergent. The limit of (s_n) is referred to as e and is used as the base for natural logarithms. The approximate value of e is 2.71828.