MATH 201 Section B
Spring 2016
Extra Credit Problems
1. For each of the following sequences, find the limit and prove that it is
correct, directly from the definition 2.1.2 of limit.
4n2 + 2
(a) xn = 2
3n − 7
√
√
(b) xn = n + 1 − n (Hint: find an equivalent expression for xn .)
sin2 (n3 )
(c) xn =
n
√
2. Let x1 = 1 and xn+1 = 1 + xn for n = 1, 2, . . . (This is a recursive
definition for the sequence. Read Section 2.2.3 for a related example.)
(a) Write out the first four terms in the sequence.
(b) Show by induction that xn ≤ 2 for all n.
(c) Show by induction that {xn }∞
n=1 is monotone increasing.
(d) It now follows by Theorem 2.1.10 that x = limn→∞ exists. Find the
value of x. (Suggestion: find an equation which x must satisfy.)