Summary: Math 2200 Problem Set 1 Key
Problem 1 Either show that the following sequences fail to converge or show they converge. When possible, compute
their limit. Hint: only one diverges CORRECTION: 2 diverge and, of the other five, four have easily computed
limits. Assume n starts large enough that zeros in the denominator are not an issue.
(i) an = Cos(2n), (ii) bn = n

n!, (iii) cn = n
n2+1 ,
(iv) dn = Cos(n·/3)

n+1
, (v) en = n2
+2n+3
(n-5)·(4-n) , and (vi) fn = n2
+1
n
Solution:
(i) This sequence is equal to one for all n so its limit is 1.
(ii) Since ln(x) is a strictly increasing function, an converges or diverges as ln(an). Notice that
ln

Source: Ashlock, Dan - Department of Mathematics and Statistics, University of Guelph

Collections: Mathematics