MATH 214 A1  Assignment 1 Solutions
Due Friday, May 13, 3:00pm
(1) Do the following sequences converge?
If so, find the limit, if
not, prove that they do not: (5 marks each)
(a)
a
n
=
3
n
4
+2
n
2
+5
n

√
n
n
4
+3
n

2
Converges: Use either l’Hopital’s rule or divide both top
and bottom by
n
4
and use limit rules.
(b)
b
n
=
n
cos
(
1
n
)
Diverges: If
b
n
converges, then
b
n
cos
(
1
n
)
=
n
converges, which
is clearly false.
(2) Suppose that
f
(
x
) is differentiable for all
x
∈
[0
,
1] and that
f
(0) = 0. If
a
n
=
nf
(
1
n
)
:
(a) Show that lim
n
→∞
a
n
=
f
0
(0)(5 marks)
By definition:
f
0
(0) = lim
h
→
0
f
(0+
h
)

f
(0)
h
= lim
h
→
0
1
h
f
(
h
).
Using
that lim
h
→
0
g
(
h
) = lim
n
→∞
g
(
1
n
) and the ”Connect the dots” the
orem from class, lim
h
→
0
1
h
f
(
h
) = lim
n
→∞
nf
(
1
n
)
(b) Use part
a
to calculate the limit of
b
n
=
n
sin
(
1
n
)
(2 marks)
Using
f
(
x
) = sin(
x
), we see that lim
h
→
0
b
n
=
f
0
(0) = cos(0) =
1.
(3) Does
∑
∞
n
=1
n
sin
(
1
n
)
converge? If so what is the value? If not,
why not? (3 marks, Hint: Look at
b
n
in question 2)
No.
lim
n
→∞
b
n
= 1
6
= 0
(4) Determine whether the following series converge or diverge. If
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 Spring '11
 AlexOndrus
 Math, Calculus, Mathematical Series, lim, converges, pn converges

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