Quiz
Sequences and series- JB 1st term revision
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Questions in this quiz (25)
1(a,b)
- Aa ≤ x ≤ b
- B{x ∈ R : a ≤ x ≤ b }
- C{x ∈ R : a < x < b }
- Da < x < b
- E(4,1)
2What are sets with curved brackets named?
- AOpen Intervals
- BClosed Intervals
3What are sets with square brackets named?
- AClosed Intervals
- BOpen Intervals
4State the triangle inequality
- A|a + b| ≤ |a| + |b|
- Ba + b ≤ |a| + |b|.
- C|a + b| < |a| + |b|
- D|a + b| ≤ |a| - |b|.
5Define what is meant by a sequence
- ACorresponds to a mapping (or ) from the natural numbers N to the real numbers R.
- BOrdered list
- CNumbers in a set
- Dcorresponds to a mapping (or ) from a number to another
- Ecorresponds to a mapping (or ) from the real numbers R to the real numbers N.
- FCorresponds to a mapping (or ) from the natural numbers N to the integers Z.
- GAn increasing list of values mapped from the Natural numbers N to the integers Z
6Define tends to infinity
- AA 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.
- BIt gets bigger and bigger past a number
- CA 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.
- DA 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.
- EA 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.
- FA 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.
7Define Tends to infinity
- A∀A>0 ∃N∈N s.t. an>A ∀n>N
- B∃A>0 ∃N∈N s.t. an>A ∀n>N
- C∀A>0 ∀N∈N s.t. anN
- D∀A>0 ∃N∈N s.t. an>A ∀n<N
8What is |x|^2 equal to ?
- Ax^2
- Bx
- C|x|
- D-x
9What's another way to write √(x^2)
- A|x|
- Bx
- Cx^2
- D|x+1|
10 |xy| =
- A|x||y|
- Bxy
- C|x+y|
- D|x|+|y|
11Define Convergent sequence
- AA 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 - BA 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 - CA 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 - DA 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 - EA 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 - FA 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 - GA 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
- AA sequence (an) of real numbers converges to a real
12(Converging series) If |an-l| = 1/n. What should you let N be greater than?
- A1/e
- Be
- C2e
- D2/e
13Define bounded above
- Aif there exists some M ∈ R such that an ≤ M for all n ∈ N
- Bif there exists some M ∈ N such that an ≤ M for all n ∈ N
- Cif there exists some M ∈ R such that an ≤ R for all n ∈ N
- Dif there exists some M ∈ R such that an ≤ M for some n ∈ N
- Eif there exists some M ∈ R such that an ≤ M for all R ∈ N
14Define bounded below
- Athere exists some M ∈ R such that an ≥ M for all n ∈ N.
- Bthere exists some M ∈ R such that an < M for all n ∈ N.
- Cthere exists some M ∈ N such that an < M for all n ∈ N.
- Dthere exists some M ∈ N such that an ≥ M for all n ∈ N.
- Ethere exists some M ∈ R such that an ≥ M for some n ∈ N.
15Define bounded
- Athere exist M1, M2 ∈ R such that M1 ≤ an ≤ M2 for all n ∈ N.
- Bthere exist M1, M2 ∈ R such that M1 ≤ an ≤ M2 for some n ∈ N.
- Cthere exist M1, M2 ∈ N such that M1 ≤ an ≤ M2 for all n ∈ N.
- Dthere exist M1, M2 ∈ Q such that M1 ≤ an ≤ M2 for all n ∈ N.
- Ethere exist M1, M2 ∈ R such that M1 < an < M2 for all n ∈ N.
- Fthere exist M1, M2 ∈ R such that M1 ≤ an < M2 for all n ∈ N.
16Give a sequence that is bounded but does not converge
an = [blank_start](-1)^n[blank_end]- A(-1)^n
17Lemma 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
- Abounded
- Bconvergent
- Creal
18AOL: lim an = ℓ and lim bn = m
Then,
lim(an + bn) = ?- Aℓ + m,
- Bℓm,
- Cℓ - m
- Dℓ + m - e
19AOL: lim an = ℓ
Then,
lim λan = ?- Aλℓ
- Bλ
- Cℓ
- D2λℓ
- Eλ+ℓ
- Fλ-ℓ
20AOL: lim an = ℓ and lim bn = m
Then,
lim anbn = ?- Aℓm
- Bℓ/m
- Cℓ + m
- Dℓ - m
21Sandwich Theorem/Squeeze Rule
- A. 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. - B. 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. - C. 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. - D. 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.
- A. Let N ∈ N and ℓ ∈ R. Suppose (an), (bn)
22If |λ| < 1 then λ^n
n → ?
as n → ∞- A0
- B1
- Cn
- D∞
- E-∞
- F2
- Gλ
23s>0
1/(n^s) → ?
as n → ∞.- A0
- B∞
- Cn
- Ds
- E1/s
24 (n^s)/ n! → ?
as n → ∞- A0
- Bn
- Cn!
- D∞
25(λ^n)/n! → ?
as n → ∞.- Aλ
- Bn!
- C0
- D∞
- En
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- Bachelors Degree Mathematics Quiz on Sequences and series- JB 1st term revision, created by Will Rickard on 26/10/2015.
- Author
- Will Rickard
- Language
- English
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