Math 312, Intro. to Real Analysis:
Midterm Exam #2
Stephen G. Simpson
Friday, March 27, 2009
1. True or False (2 points each)
(a) Every monotone sequence of real numbers is convergent.
(b) Every sequence of real numbers has a limsup and a lim inf .
(c) Every sequence of real numbers has a monotone subsequence.
(d) Every sequence of real numbers has a convergent subsequence.
(e) If lim inf an= lim sup an=αthen lim an=α.
(f) We can find a sequence of real numbers, (an), such that the sub-
sequential limits of (an) are exactly the real numbers in the closed
interval [−1,1].
(g) The series 1 −1
4+1
9−1
16 +···is absolutely convergent.
(h) If P|an|is convergent, then so is P|an|2.
(i) If Panis convergent, then so is Pa2
n.
(j)
∞
X
n=1
1
npis convergent for all p≥1.
(k)
∞
X
n=1
1
10n=1
9.
(l) The function √xis uniformly continuous on [0,∞).
(m) If a function is uniformly continuous on the interval (a, b] and on the
interval [b,c), then it is uniformly continuous on the interval (a, c).
2. (6 points each) Which of the following series are convergent and/or ab-
solutely convergent? Please indicate which tests you are using and show
your work.
(a) 1 −1
2+1
3−1
4+···
(b)
∞
X
n=1
(pn2+n−n)
(c)
∞
X
n=1
(pn2+ 1 −n)
(d) X1
1.01n−1000
(e) X1
n2+√n2+ 1
(f)
∞
X
n=1
log n+ 1
n
1
