myquiz1s
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Math3068 Analysis. Name: (Bring photo ID so the lecturer can verify this)
Quiz 1, 2009 Student Number: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Sample) Signature: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The quiz will be marked out of 5, with 5 questions worth 1 each (unlike this sample quiz has morethan 5 questions). Write your answers clearly, and in the boxes provided. Illegible answers will be
counted as wrong.
1. Find the limit ℓ of the sequence whose n-th term is xn = 2n+n
3n−2n+7 .
ℓ =
2. Find the limit ℓ of the sequence whose n-th term is xn =
1 + 3
n
n.
ℓ =
3. You are given that the sequence (xn) defined by setting x0 = 0 and xn+1 =√
3 + xn for all n ≥ 0converges to a limit ℓ. Find ℓ.
ℓ =
4. Is the following statement true or false? If it is false, then give a counterexample. “If an → 0,then the series
∞k=0
ak converges.”
5. Find the sum s of the series ∞
k=1
1√ k+1
− 1√ k+2
.
s =
6. Assume that |z| < 1. Find the sum s of the series 1 − z3 + z6 − z9 + z12 − · · · .
s =
7. Find the radius of convergence R of the power series ∞
k=0k
3kz2k.
R =
8. Evaluate the binomial coefficient−33
.
−33
=
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9. We can expand e2z in a power series a0 + a1z + a2z2 + · · · . Write down the coefficient a4 of z4.
a4 =
Answers:1: ℓ = 0.2: ℓ = e3.3: ℓ = 1+
√ 13
2 .
4: False. For example an = 1/n, but ∞
k=11
k diverges.
5: s = 1√ 2
.
6: 1
1+z3.
7: R =√
3.8:
−33
= −10.
9: a4 = 2
3.
2