Illustrative Example 2. (B ) found in the last example, calculate
s
i
n
1
{\displaystyle sin1}
Solution. Here
x
=
1
{\displaystyle x=1}
x
=
1
{\displaystyle x=1}
(B ) of the last example,
1
−
1
3
!
+
1
5
!
−
1
7
!
+
⋯
.
{\displaystyle 1-{\tfrac {1}{3!}}+{\tfrac {1}{5!}}-{\tfrac {1}{7!}}+\cdots .}
Summing up the positive and negative terms separately,
1
=
1.00000
⋯
1
3
!
=
0.16667
⋯
=
0.00833
⋯
1
7
!
=
0.00019
⋯
1.00833
⋯
0.16686
⋯
{\displaystyle {\begin{array}{rclrcl}1&=&1.00000\cdots &\qquad {\frac {1}{3!}}&=&0.16667\cdots \\&=&0.00833\cdots &\qquad {\frac {1}{7!}}&=&0.00019\cdots \\\hline &&1.00833\cdots &&&0.16686\cdots \\\end{array}}}
Hence
sin
1
=
1.00833
−
0.16686
=
0.84147
⋯
,
{\displaystyle \sin 1=1.00833-0.16686=0.84147\cdots ,}
which is correct to four decimal places, since the error made must be less than;
1
9
!
;
{\displaystyle {\tfrac {1}{9!}};}
sin
1
{\displaystyle \sin 1}
Verify the following expansions of functions into power series by Maclaurin's Series and determine for what values of the variable they are convergent:
1.
e
x
=
1
+
x
+
x
2
2
!
+
x
3
3
!
+
x
4
4
!
+
⋯
.
{\displaystyle e^{x}=1+x+{\tfrac {x^{2}}{2!}}+{\tfrac {x^{3}}{3!}}+{\tfrac {x^{4}}{4!}}+\cdots .}
Convergent for all values of
x
{\displaystyle x}
2.
cos
x
=
1
−
x
2
2
!
+
x
4
4
!
−
x
6
6
!
+
x
8
8
!
−
⋯
.
{\displaystyle \cos x=1-{\tfrac {x^{2}}{2!}}+{\tfrac {x^{4}}{4!}}-{\tfrac {x^{6}}{6!}}+{\tfrac {x^{8}}{8!}}-\cdots .}
Convergent for all values of
x
{\displaystyle x}
3.
a
x
=
1
+
x
log
a
+
x
2
log
2
a
2
!
+
x
3
log
3
a
3
!
+
⋯
.
{\displaystyle a^{x}=1+x\log a+{\frac {x^{2}\log ^{2}a}{2!}}+{\frac {x^{3}\log ^{3}a}{3!}}+\cdots .}
Convergent for all values of
x
{\displaystyle x}
4.
sin
k
x
=
k
x
−
k
3
x
3
3
!
+
k
5
x
5
5
!
−
k
7
x
7
7
!
+
⋯
.
{\displaystyle \sin kx=kx-{\frac {k^{3}x^{3}}{3!}}+{\frac {k^{5}x^{5}}{5!}}-{\frac {k^{7}x^{7}}{7!}}+\cdots .}
Convergent for all values of
x
{\displaystyle x}
k
{\displaystyle k}
5.
e
−
k
x
=
1
−
k
x
+
k
2
x
2
2
!
−
k
3
x
3
3
!
+
k
4
x
4
4
!
+
⋯
.
{\displaystyle e^{-kx}=1-kx+{\frac {k^{2}x^{2}}{2!}}-{\frac {k^{3}x^{3}}{3!}}+{\frac {k^{4}x^{4}}{4!}}+\cdots .}
Convergent for all values of
x
{\displaystyle x}
k
{\displaystyle k}
6.
log
(
1
+
x
)
=
x
−
x
2
2
+
x
3
3
−
x
4
4
+
x
5
5
−
⋯
.
{\displaystyle \log \left(1+x\right)=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+{\frac {x^{5}}{5}}-\cdots .}
Convergent if
−
1
<
x
≤
1
{\displaystyle -1<x\leq 1}
7.
log
(
1
−
x
)
=
x
−
x
2
2
−
x
3
3
−
x
4
4
−
x
5
5
−
⋯
.
{\displaystyle \log \left(1-x\right)=x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}-{\frac {x^{5}}{5}}-\cdots .}
Convergent if
−
1
<
x
≤
1
{\displaystyle -1<x\leq 1}
8.
arcsin
x
=
x
+
1
⋅
x
3
2
⋅
3
+
1
⋅
3
x
5
2
⋅
4
⋅
5
+
⋯
.
{\displaystyle \arcsin x=x+{\frac {1\cdot x^{3}}{2\cdot 3}}+{\frac {1\cdot 3x^{5}}{2\cdot 4\cdot 5}}+\cdots .}
Convergent if
−
1
≤
x
≤
1
{\displaystyle -1\leq x\leq 1}
9.
arctan
x
=
x
−
x
3
3
+
x
5
5
−
x
7
7
+
x
9
9
−
⋯
.
{\displaystyle \arctan x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+{\frac {x^{9}}{9}}-\cdots .}
Convergent if
−
1
≤
x
≤
1
{\displaystyle -1\leq x\leq 1}
10.
sin
x
2
=
x
2
−
2
x
4
3
!
+
32
x
6
6
!
+
⋯
.
{\displaystyle \sin x^{2}=x^{2}-{\frac {2x^{4}}{3!}}+{\frac {32x^{6}}{6!}}+\cdots .}
Convergent if
−
1
≤
x
≤
1
{\displaystyle -1\leq x\leq 1}
11.
e
sin
ϕ
=
1
+
ϕ
+
ϕ
2
2
−
ϕ
4
8
+
⋯
.
{\displaystyle e^{\sin \phi }=1+\phi +{\frac {\phi ^{2}}{2}}-{\frac {\phi ^{4}}{8}}+\cdots .}
Convergent for all values of
ϕ
{\displaystyle \phi }
12.
e
θ
sin
θ
=
θ
+
θ
2
+
θ
3
3
−
4
θ
5
5
!
−
8
θ
6
6
!
−
⋯
.
{\displaystyle e^{\theta }\sin \theta =\theta +\theta ^{2}+{\frac {\theta ^{3}}{3}}-{\frac {4\theta ^{5}}{5!}}-{\frac {8\theta ^{6}}{6!}}-\cdots .}
Convergent for all values of
θ
{\displaystyle \theta }
