taylor series polynomials (common)
DESCRIPTION
Taylor ExpansionsTRANSCRIPT
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Commonly Used Taylor Series
series when is valid/true
11 x = 1 + x + x
2 + x3 + x4 + . . .note this is the geometric series.
just think of x as r
=n=0
xn x (1, 1)
ex = 1 + x +x2
2!+
x3
3!+
x4
4!+ . . .
so:e = 1 + 1 + 12! +
13! +
14! + . . .
e(17x) =
n=0(17x)n
n! =n=0
17nxn
n!
=n=0
xn
n!x R
cosx = 1 x2
2!+
x4
4! x
6
6!+
x8
8! . . .
note y = cosx is an even function(i.e., cos(x) = +cos(x)) and thetaylor seris of y = cosx has only
even powers.
=n=0
(1)n x2n
(2n)!x R
sinx = x x3
3!+
x5
5! x
7
7!+
x9
9! . . .
note y = sinx is an odd function(i.e., sin(x) = sin(x)) and thetaylor seris of y = sinx has only
odd powers.
=n=1
(1)(n1) x2n1
(2n 1)!or=
n=0
(1)n x2n+1
(2n + 1)!x R
ln (1 + x) = x x2
2+
x3
3 x
4
4+
x5
5 . . . question: is y = ln(1 + x) even,
odd, or neither?
=n=1
(1)(n1) xn
n
or=n=1
(1)n+1xn
nx (1, 1]
tan1 x = x x3
3+
x5
5 x
7
7+
x9
9 . . . question: is y = arctan(x) even,
odd, or neither?
=n=1
(1)(n1) x2n1
2n 1or=
n=0
(1)n x2n+1
2n + 1x [1, 1]
1