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Sequences and series- JB 1st term revision

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Bachelors Degree Mathematics Quiz on Sequences and series- JB 1st term revision, created by Will Rickard on 26/10/2015.
Will Rickard
Quiz by Will Rickard, updated more than 1 year ago
Will Rickard
Created by Will Rickard over 8 years ago
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Resource summary

Question 1

Question
(a,b)
Answer
  • a ≤ x ≤ b
  • {x ∈ R : a ≤ x ≤ b }
  • {x ∈ R : a < x < b }
  • a < x < b
  • (4,1)

Question 2

Question
What are sets with curved brackets named?
Answer
  • Open Intervals
  • Closed Intervals

Question 3

Question
What are sets with square brackets named?
Answer
  • Closed Intervals
  • Open Intervals

Question 4

Question
State the triangle inequality
Answer
  • |a + b| ≤ |a| + |b|
  • a + b ≤ |a| + |b|.
  • |a + b| < |a| + |b|
  • |a + b| ≤ |a| - |b|.

Question 5

Question
Define what is meant by a sequence
Answer
  • Corresponds to a mapping (or ) from the natural numbers N to the real numbers R.
  • Ordered list
  • Numbers in a set
  • corresponds to a mapping (or ) from a number to another
  • corresponds to a mapping (or ) from the real numbers R to the real numbers N.
  • Corresponds to a mapping (or ) from the natural numbers N to the integers Z.
  • An increasing list of values mapped from the Natural numbers N to the integers Z

Question 6

Question
Define tends to infinity
Answer
  • A sequence (an) of real numbers tends to infinity if given any real number A > 0 there exists N ∈ N such that an>A for all n>N.
  • It gets bigger and bigger past a number
  • A sequence (an) of numbers tends to infinity if given any number A > 0 there exists N ∈ N such that an>A for all n>N.
  • A sequence (an) of real numbers goes to infinity if given any real number A > 0 there exists N ∈ N such that an>A for all A>N.
  • A sequence (an) of real numbers tends to infinity if given any real number A > 0 there exists N ∈ N such that an>A for some A>N.
  • A sequence (an) of real numbers tends to infinity if given any real number A > 0 there exists z ∈ Z such that an>A for all A>N.

Question 7

Question
Define Tends to infinity
Answer
  • ∀A>0 ∃N∈N s.t. an>A ∀n>N
  • ∃A>0 ∃N∈N s.t. an>A ∀n>N
  • ∀A>0 ∀N∈N s.t. anN
  • ∀A>0 ∃N∈N s.t. an>A ∀n<N

Question 8

Question
What is |x|^2 equal to ?
Answer
  • x^2
  • x
  • |x|
  • -x

Question 9

Question
What's another way to write √(x^2)
Answer
  • |x|
  • x
  • x^2
  • |x+1|

Question 10

Question
|xy| =
Answer
  • |x||y|
  • xy
  • |x+y|
  • |x|+|y|

Question 11

Question
Define Convergent sequence
Answer
  • A sequence (an) of real numbers converges to a real number ℓ if given any e > 0 there exists N ∈ N such that |an − ℓ| < e for all n > N
  • A sequence (an) of numbers converges to a real number ℓ if given any e > 0 there exists N ∈ N such that |an − ℓ| < e for all n > N
  • A sequence (an) of real numbers converges to a number ℓ if given any e > 0 there exists N ∈ N such that |an − ℓ| < e for all n > N
  • A sequence (an) of real numbers converges to a real number ℓ if given any e < 0 there exists N ∈ N such that |an − ℓ| < e for all n > N
  • A sequence (an) of real numbers converges to a real number ℓ if given any e > 0 there exists Z ∈ N such that |an − ℓ| < e for all n > N
  • A sequence (an) of real numbers converges to a real number ℓ if given any e > 0 there exists N ∈ N such that |an − ℓ| < e for some n > N
  • A sequence (an) of real numbers converges to a real number ℓ if given any e > 0 there exists N ∈ N such that |e − ℓ| < e for all n > N

Question 12

Question
(Converging series) If |an-l| = 1/n. What should you let N be greater than?
Answer
  • 1/e
  • e
  • 2e
  • 2/e

Question 13

Question
Define bounded above
Answer
  • if there exists some M ∈ R such that an ≤ M for all n ∈ N
  • if there exists some M ∈ N such that an ≤ M for all n ∈ N
  • if there exists some M ∈ R such that an ≤ R for all n ∈ N
  • if there exists some M ∈ R such that an ≤ M for some n ∈ N
  • if there exists some M ∈ R such that an ≤ M for all R ∈ N

Question 14

Question
Define bounded below
Answer
  • there exists some M ∈ R such that an ≥ M for all n ∈ N.
  • there exists some M ∈ R such that an < M for all n ∈ N.
  • there exists some M ∈ N such that an < M for all n ∈ N.
  • there exists some M ∈ N such that an ≥ M for all n ∈ N.
  • there exists some M ∈ R such that an ≥ M for some n ∈ N.

Question 15

Question
Define bounded
Answer
  • there exist M1, M2 ∈ R such that M1 ≤ an ≤ M2 for all n ∈ N.
  • there exist M1, M2 ∈ R such that M1 ≤ an ≤ M2 for some n ∈ N.
  • there exist M1, M2 ∈ N such that M1 ≤ an ≤ M2 for all n ∈ N.
  • there exist M1, M2 ∈ Q such that M1 ≤ an ≤ M2 for all n ∈ N.
  • there exist M1, M2 ∈ R such that M1 < an < M2 for all n ∈ N.
  • there exist M1, M2 ∈ R such that M1 ≤ an < M2 for all n ∈ N.

Question 16

Question
Give a sequence that is bounded but does not converge an = [blank_start](-1)^n[blank_end]
Answer
  • (-1)^n

Question 17

Question
Lemma 1.9, Convergent sequences are bounded. Every [blank_start]convergent[blank_end] sequence of [blank_start]real[blank_end] numbers is a [blank_start]bounded[blank_end] sequence
Answer
  • bounded
  • convergent
  • real

Question 18

Question
AOL: lim an = ℓ and lim bn = m Then, lim(an + bn) = ?
Answer
  • ℓ + m,
  • ℓm,
  • ℓ - m
  • ℓ + m - e

Question 19

Question
AOL: lim an = ℓ Then, lim λan = ?
Answer
  • λℓ
  • λ
  • ℓ
  • 2λℓ
  • λ+ℓ
  • λ-ℓ

Question 20

Question
AOL: lim an = ℓ and lim bn = m Then, lim anbn = ?
Answer
  • ℓm
  • ℓ/m
  • ℓ + m
  • ℓ - m

Question 21

Question
Sandwich Theorem/Squeeze Rule
Answer
  • . Let N ∈ N and ℓ ∈ R. Suppose (an), (bn) and (cn) are sequences satisfying an ≤ bn ≤ cn for all n ≥ N. If an → ℓ and cn → ℓ, then bn → ℓ.
  • . Let N ∈ N and ℓ ∈ R. Suppose (an), (bn) and (cn) are sequences satisfying an ≤ n ≤ cn for all n ≥ N. If an → ℓ and cn → ℓ, then bn → ℓ.
  • . Let N ∈ R and ℓ ∈ N. Suppose (an), (bn) and (cn) are sequences satisfying an ≤ bn ≤ cn for all n ≥ N. If an → ℓ and cn → ℓ, then bn → ℓ.
  • . Let N ∈ N and ℓ ∈ R. Suppose (an), (bn) and (cn) are sequences satisfying an ≤ bn ≤ cn for some n ≥ N. If an → ℓ and cn → ℓ, then bn → ℓ.

Question 22

Question
If |λ| < 1 then λ^n n → ? as n → ∞
Answer
  • 0
  • 1
  • n
  • ∞
  • -∞
  • 2
  • λ

Question 23

Question
s>0 1/(n^s) → ? as n → ∞.
Answer
  • 0
  • ∞
  • n
  • s
  • 1/s

Question 24

Question
(n^s)/ n! → ? as n → ∞
Answer
  • 0
  • n
  • n!
  • ∞

Question 25

Question
(λ^n)/n! → ? as n → ∞.
Answer
  • λ
  • n!
  • 0
  • ∞
  • n
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