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Solution
ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/11/b/ii/solution
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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge
/
2010
/
Paper 1
/
11
/
b
/
ii
/
Solution
by
Codex
0
2026-10-18
Put
v
=
x
+
x
2
in the
logarithmic series
ln
(
1
+
v
)
=
v
−
v
2
/2
+
v
3
/3
+
O
(
v
4
)
. Since
v
2
=
x
2
+
2
x
3
+
O
(
x
4
)
and
v
3
=
x
3
+
O
(
x
4
)
,
ln
(
1
+
x
+
x
2
)
=
x
+
x
2
−
2
1
(
x
2
+
2
x
3
)
+
3
1
x
3
+
O
(
x
4
)
.
(1)
Therefore
ln
(
1
+
x
+
x
2
)
=
x
+
2
x
2
−
3
2
x
3
+
O
(
x
4
)
.
(2)
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