7/24/2019 myquiz1s

http://slidepdf.com/reader/full/myquiz1s 1/2

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

 =

Copyright   c 2009 The University of Sydney   1

7/24/2019 myquiz1s

http://slidepdf.com/reader/full/myquiz1s 2/2

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