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Material Type: Exam; Professor: Maschler; Class: INTRO TO ANALYSIS; Subject: Mathematics; University: Clark University; Term: Fall 2010;
Typology: Exams
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MATH 172 - Introduction to Analysis Midterm, Fall 2010
Instructor: Gideon Maschler Total: 100 marks Answer all questions clearly, and with complete justifications. 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
ํ=
(2ํ โ 1) = ํ^2 holds for the sum of
the first ํ odd natural numbers. [14] 2) a) For ํ, ํ, ํ, ํ โ โ, state the condition under which the pairs (ํ, ํ) and (ํ, ํ) represent the same integer in โค (i.e. when does (ํ, ํ) = (ํ, ํ) hold?) b) Show that the negation of integers is well-defined, i.e. if (ํ, ํ) = (ํ, ํ), then โ(ํ, ํ) = โ(ํ, ํ).
[18] 3) Suppose lim ํโโ ํํ = ํ and lim ํโโ ํํ = ํ. Show that lim ํโโ
[18] 4) Suppose that {ํํ}โ ํ=1 is a bounded sequence, and lim ํโโ ํํ = 0. Show that lim ํโโ
[18] 5) Given an ํ-ํ proof that lim ํโโ
[18] 6) a) Find the sum for
ํ=
. (Hint: (^) ํ(ํ^2 +2) = (^) ํ^1 โ (^) ํ^1 +2 .)
b) Determine whether
ํ=
converges.
c) Determine whether
ํ=
converges.
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