13.
(a)
tan
x
.
{\displaystyle \tan x.}
(b)
sec
x
.
{\displaystyle \sec x.}
(c)
e
cos
x
.
{\displaystyle e^{\cos x}.}
(d)
cos
2
x
.
{\displaystyle \cos 2x.}
(e)
arccos
x
.
{\displaystyle \arccos x.}
(f)
a
−
x
.
{\displaystyle a^{-x}.}
14.
log
x
{\displaystyle \log x}
Compute the values of the following functions by substituting directly in the equivalent power series, taking terms enough until the results agree with those given below.
15
e
=
2.7182
⋯
.
{\displaystyle e=2.7182\cdots .}
Solution. Let
x
=
1
{\displaystyle x=1}
Ex. 1
e
=
1
+
1
1
2
!
+
1
3
!
+
1
4
!
+
1
5
!
+
⋯
.
{\displaystyle e=1+1{\tfrac {1}{2!}}+{\tfrac {1}{3!}}+{\tfrac {1}{4!}}+{\tfrac {1}{5!}}+\cdots .}
First term
=
1.00000
{\displaystyle =1.00000}
Second term
=
1.00000
{\displaystyle =1.00000}
Third term
=
0.50000
{\displaystyle =0.50000}
Fourth term
=
0.16667
⋯
{\displaystyle =0.16667\cdots }
(Dividing third term by 3.)
Fifth term
=
0.04167
⋯
{\displaystyle =0.04167\cdots }
(Dividing fourth term by 4.)
Sixth term
=
0.00833
⋯
{\displaystyle =0.00833\cdots }
(Dividing fifth term by 5.)
Seventh term
=
0.00139
⋯
{\displaystyle =0.00139\cdots }
(Dividing sixth term by 6.)
Eighth term
=
0.00019
⋯
,
{\displaystyle =0.00019\cdots ,}
(Divding sevent term by 7.)
Adding,
e
{\displaystyle e}
=
2.71825
⋯
{\displaystyle =2.71825\cdots }
Ans.
16.
arctan
(
1
5
)
=
0.1973
⋯
;
{\displaystyle \arctan \left({\tfrac {1}{5}}\right)=0.1973\cdots ;}
Ex. 9
17.
cos
1
=
0.5403
⋯
;
{\displaystyle \cos 1=0.5403\cdots ;}
Ex. 2
18.
cos
10
∘
=
0.9848
⋯
;
{\displaystyle \cos 10^{\circ }=0.9848\cdots ;}
Ex. 2
19.
sin
.1
=
.0998
⋯
;
{\displaystyle \sin .1=.0998\cdots ;}
x
−
x
3
3
!
+
x
5
5
!
−
x
7
7
!
+
⋯
.
{\displaystyle x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots .}
20.
arcsin
1
=
1.5708
⋯
;
{\displaystyle \arcsin 1=1.5708\cdots ;}
Ex. 8
21.
sin
π
4
=
0.7071
⋯
;
{\displaystyle \sin {\tfrac {\pi }{4}}=0.7071\cdots ;}
(B ), §145 .
22.
sin
.5
=
0.4794
⋯
;
{\displaystyle \sin .5=0.4794\cdots ;}
(B ), §145 .
23.
e
2
=
1
+
2
+
2
2
2
!
+
2
3
3
!
+
⋯
=
7.3891.
{\displaystyle e^{2}=1+2+{\tfrac {2^{2}}{2!}}+{\tfrac {2^{3}}{3!}}+\cdots =7.3891.}
24.
e
=
1
1
2
+
1
2
2
(
2
!
)
+
1
2
3
(
3
!
)
+
⋯
=
1.6487.
{\displaystyle {\sqrt {e}}=1{\frac {1}{2}}+{\frac {1}{2^{2}\left(2!\right)}}+{\frac {1}{2^{3}\left(3!\right)}}+\cdots =1.6487.}
In more advanced treatises it is shown that, for values of
x
{\displaystyle x}
(B ), §145 , Differentiating both sides,
sin
x
=
x
−
x
3
3
!
+
x
5
5
!
−
x
7
7
!
+
⋯
.
{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots .}
Differentiating both sides, we get
cos
x
=
1
−
x
2
2
!
+
x
4
4
!
−
x
6
6
!
+
⋯
,
{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots ,}
