Math 4700 UMC Advanced Calculus Worksheet

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1.  (5 pts) Prove using the definition of the limit (findN(?) for every?) thatlimn??2n2+ 13n2?1=23.
2.  (5 pts) Which of the following statements imply that the sequence{xn}is bounded?  Prove your answer.
(A) There exist? >0 anda?Rso that for everyn?Nsatisfyingn >100?we have|xn?a|< ?. (B) There existsa?Rsuch that for every? >0 we have|xn?a|< ?for alln?Nsatisfyingn <100?. (C) There exist? >0 anda?Rso that we have|xn?a|< ?foreveryn?Nsatisfyingn <100?. (D) There exists? >0 such that for everya?(??,?10)?(10,?)we have|xn?a|> ?for everyn?Nsatisfyingn >100?.

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QUIZ 2. ADVANCED CALCULUS 1, FALL 2020
1. (5 pts) Prove using the definition of the limit (find N (ε) for every
ε) that
2n2 + 1
2
lim
= .
2
n→∞ 3n − 1
3
2. (5 pts) Which of the following statements imply that the sequence
{xn } is bounded? Prove your answer.
(A) There exist ε > 0 and a ∈ R so that for every n ∈ N satisfying
n > 100
we have |xn − a| < ε. ε (B) There exists a ∈ R such that for every ε > 0 we have |xn − a| < ε for all n ∈ N satisfying n < 100 . ε (C) There exist ε > 0 and a ∈ R so that we have |xn − a| < ε for . every n ∈ N satisfying n < 100 ε (D) There exists ε > 0 such that for every a ∈ (−∞, −10) ∪ (10, ∞)
we have |xn − a| > ε for every n ∈ N satisfying n > 100
.
ε
3. (5 pts) Suppose that two sequences {xn } and {yn } converge to the
same limit a ∈ R. Define a sequence {zn } by z2n = xn , z2n−1 = −yn
for all n ∈ N. For which numbers a does the sequence {zn } converge?
Prove your answer.
1

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mathematics

calculus

Advanced calculus

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