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Midterm Exam - Introduction to Analysis - Fall 2010 | MATH 172, Exams of Mathematical Methods for Numerical Analysis and Optimization

Material Type: Exam; Professor: Maschler; Class: INTRO TO ANALYSIS; Subject: Mathematics; University: Clark University; Term: Fall 2010;

Typology: Exams

2009/2010

Uploaded on 12/14/2010

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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 justi๏ฌcations. 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‘›
โˆ‘
๐‘˜=1
(2๐‘˜โˆ’1) = ๐‘›2holds for the sum of
the ๏ฌrst ๐‘›odd natural numbers.
[14] 2) a) For ๐‘Ž, ๐‘, ๐‘, ๐‘‘ โˆˆโ„•, state the condition under which the pairs (๐‘Ž, ๐‘) and
(๐‘, ๐‘‘) represent the same integer in โ„ค(i.e. when does (๐‘Ž, ๐‘) = (๐‘, ๐‘‘)
hold?)
b) Show that the negation of integers is well-de๏ฌned, i.e. if (๐‘Ž, ๐‘) = (๐‘, ๐‘‘),
then โˆ’(๐‘Ž, ๐‘) = โˆ’(๐‘, ๐‘‘).
[18] 3) Suppose lim
๐‘›โ†’โˆž
๐‘Ž๐‘›=๐‘Žand lim
๐‘›โ†’โˆž
๐‘๐‘›=๐‘. Show that lim
๐‘›โ†’โˆž
(๐‘Ž๐‘›โˆ’๐‘๐‘›) = ๐‘Žโˆ’๐‘.
[18] 4) Suppose that {๐‘Ž๐‘›}โˆž
๐‘›=1 is a bounded sequence, and lim
๐‘›โ†’โˆž
๐‘๐‘›= 0. Show that
lim
๐‘›โ†’โˆž
๐‘Ž๐‘›๐‘๐‘›= 0.
[18] 5) Given an ๐œ–-๐‘proof that lim
๐‘›โ†’โˆž
3๐‘›+ 2
2๐‘›+ 1 =3
2.
[18] 6) a) Find the sum for
โˆž
โˆ‘
๐‘›=1
2
๐‘›(๐‘›+ 2). (Hint: 2
๐‘›(๐‘›+2) =1
๐‘›โˆ’1
๐‘›+2 .)
b) Determine whether
โˆž
โˆ‘
๐‘›=1
1
โˆš๐‘›3+ 4 converges.
c) Determine whether
โˆž
โˆ‘
๐‘›=1
๐‘›
3๐‘›converges.
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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

X^ ํ‘›

ํ‘˜=

(2ํ‘˜ โˆ’ 1) = ํ‘›^2 holds for the sum of

the first ํ‘› odd natural numbers. [14] 2) a) For ํ‘Ž, ํ‘, ํ‘, ํ‘‘ โˆˆ โ„•, state the condition under which the pairs (ํ‘Ž, ํ‘) and (ํ‘, ํ‘‘) represent the same integer in โ„ค (i.e. when does (ํ‘Ž, ํ‘) = (ํ‘, ํ‘‘) hold?) b) Show that the negation of integers is well-defined, i.e. if (ํ‘Ž, ํ‘) = (ํ‘, ํ‘‘), then โˆ’(ํ‘Ž, ํ‘) = โˆ’(ํ‘, ํ‘‘).

[18] 3) Suppose lim ํ‘›โ†’โˆž ํ‘Žํ‘› = ํ‘Ž and lim ํ‘›โ†’โˆž ํ‘ํ‘› = ํ‘. Show that lim ํ‘›โ†’โˆž

[18] 4) Suppose that {ํ‘Žํ‘›}โˆž ํ‘›=1 is a bounded sequence, and lim ํ‘›โ†’โˆž ํ‘ํ‘› = 0. Show that lim ํ‘›โ†’โˆž

[18] 5) Given an ํœ–-ํ‘ proof that lim ํ‘›โ†’โˆž

[18] 6) a) Find the sum for

X^ โˆž

ํ‘›=

. (Hint: (^) ํ‘›(ํ‘›^2 +2) = (^) ํ‘›^1 โˆ’ (^) ํ‘›^1 +2 .)

b) Determine whether

X^ โˆž

ํ‘›=

ํ‘›^3 + 4

converges.

c) Determine whether

X^ โˆž

ํ‘›=

3 ํ‘›^

converges.

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