MATH 172 - Introduction to Analysis
Midterm, Fall 2010
Instructor: Gideon Maschler Total: 100 marks
Answer all questions clearly, and with complete justi๏ฌcations. Be precise in quoting
material which you use that was learned in class. There are 6 problems, the value
of each is indicated by [ ].
[14] 1) Prove by induction that the formula
๐
โ
๐=1
(2๐โ1) = ๐2holds for the sum of
the ๏ฌrst ๐odd natural numbers.
[14] 2) a) For ๐, ๐, ๐, ๐ โโ, state the condition under which the pairs (๐, ๐) and
(๐, ๐) represent the same integer in โค(i.e. when does (๐, ๐) = (๐, ๐)
hold?)
b) Show that the negation of integers is well-de๏ฌned, i.e. if (๐, ๐) = (๐, ๐),
then โ(๐, ๐) = โ(๐, ๐).
[18] 3) Suppose lim
๐โโ
๐๐=๐and lim
๐โโ
๐๐=๐. Show that lim
๐โโ
(๐๐โ๐๐) = ๐โ๐.
[18] 4) Suppose that {๐๐}โ
๐=1 is a bounded sequence, and lim
๐โโ
๐๐= 0. Show that
lim
๐โโ
๐๐๐๐= 0.
[18] 5) Given an ๐-๐proof that lim
๐โโ
3๐+ 2
2๐+ 1 =3
2.
[18] 6) a) Find the sum for
โ
โ
๐=1
2
๐(๐+ 2). (Hint: 2
๐(๐+2) =1
๐โ1
๐+2 .)
b) Determine whether
โ
โ
๐=1
1
โ๐3+ 4 converges.
c) Determine whether
โ
โ
๐=1
๐
3๐converges.
1