Course: Accelerated Engineering Calculus II
Instructor: Michael Medvinsky
Worksheet 1
1. Write the following sequence as { an }n=1
¥
a.
1 3 5 7
, , , ...
2 4 6 8
¥
ì1ü
1 1 1 1
b. , , , ... = í n ý
3 9 27 81
î 3 þn=1
2. Find following limits
a. lim ( -1)
n+1
n®¥
=
æ æ 1ö nö
b. lim ç 1+ ç - ÷ ÷ =
n®¥ ç
è è 2 ø ÷ø
c. lim n +1 =
n®¥
d. lim
n®¥
ln n
=
n
3. Give a function f ( x ) and a sequence an find lim f ( an )
n®¥
a. f ( x ) = sin x; an =
p
1
x
1
n
b. f ( x ) = ;
an =
æ 1ö
+ç- n ÷
2 è 2 ø
Course: Accelerated Engineering Calculus II
4. Let an =
Instructor: Michael Medvinsky
n
n +1
a. Determine whether the sequence is bounded or not.
b. Does it convergent or divergent, why? (don’t use the limit here)
c. Find lim
n®¥
n
=
n +1
5. Check monotonicity of
3 + 5n 2
n + n2
6. Find limit of recursive sequence: a1 = 10 , an+1 = ( 2 + an ) / 2
Hint:
L = lim an = lim an+1
n®¥
7. lim (1+ 6n ) =
n
n®¥
n®¥