Basic Maclaurin Series
sin x=x−
x3
3! +x5
5! −
x7
7! +··· =
∞
X
n=0
(−1)nx2n+1
(2n+ 1)! −∞ < x < +∞
cos x= 1 −
x2
2! +x4
4! −
x6
6! +··· =
∞
X
n=0
(−1)nx2n
(2n)! −∞ < x < +∞
ex= 1 + x+x2
2! +x3
3! +x4
4! +··· =
∞
X
n=0
xn
n!−∞ < x < +∞
sinh x=ex−e−x
2=x+x3
3! +x5
5! +x7
7! +··· =
∞
X
n=0
x2n+1
(2n+ 1)! −∞ < x < +∞
cosh x=ex+e−x
2= 1 + x2
2! +x4
4! +x6
6! +··· =
∞
X
n=0
x2n
(2n)! −∞ < x < +∞
1
1−x= 1 + x+x2+x3+x4+· ·· =
∞
X
n=0
xn−1< x < 1
ln(1 + x) = x−
x2
2+x3
3−
x4
4+··· =
∞
X
n=1
(−1)n+1xn
n−1< x ≤1
arctan x=x−
x3
3+x5
5−
x7
7+··· =
∞
X
n=0
(−1)nx2n+1
2n+ 1 −1≤x≤1
(1 + x)k= 1 + kx +k(k−1)x2
2! +k(k−1)(k−2)x3
3! +··· −1< x < 1
√1 + x= (1 + x)1
2= 1 + x
2−
x2
8+x3
16 − · ·· −1< x < 1
arcsin x=x+x3
2·3+1·3x5
2·4·5+1·3·5x7
2·4·6·7+··· =
∞
X
n=0
(2n)!x2n+1
(2nn!)2(2n+ 1) −1< x < 1