MATH 120
RECITATION QUESTIONS
WEEK 4
1. Find the Maclaurin series for
f
(
x
) = (1 +
x
)

3
, and the radius and interval of convergence,
also.
2. Find the Maclaurin series of
f
(
x
) = cos
x
2
and its radius of convergence. Compare the graph
of
f
and those of its ﬁrst three Taylor polynomials.
3. Use series to approximate the deﬁnite integral
Z
1
0
x
cos(
x
3
)
dx
correct to three decimal places.
4. Use series to evaluate the limits
a
) lim
x
→
0
sin
x

x
+
1
6
x
3
x
5
; [Ans:1/120]
b
) lim
x
→
0
sin(sin
x
)

x
x
[cos(sin
x
)

1]
; [Ans:2/3]
5. Use the binomial series to expand the function
f
(
x
) =
x
√
4 +
x
2
as a power series. State the
radius of convergence.
6. Try to attempt L’Hospital Rule to evaluate the limit
lim
x
→
0
(
e
2
x

1) ln(1 +
x
3
)
(1

cos 3
x
)
2
and compare the evaluation (of this limit) using series. [Ans:8/81]
7. Expand
f
(
x
) =
x
+
x
2
(1

x
)
3
as a power series and use this to ﬁnd the sum of the series
∞
X
n
=1
n
2
2
n
.
[Ans:6]
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 Spring '10
 Tor
 Maclaurin Series, Taylor Series, series A

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