If a function can be expressed in the form of,
f(x) = f(a) + f'(a)(x-a) + f"(a)(x-a)2/2! + ...,
it is considered as Taylor's Series.
Taylor Series: f(x) = f(a) + f'(a)(x-a) + f"(a)(x-a)2/2! + ...
If a = 0, then it leads to another series, known as Maclaurin Series
So, Maclaurin Series is as follows:
Maclaurin Series: f(x) = f(0) + f'(0)x/1! + f"(0)x2/2!....
Taylor series leads to the following power series:
If f(x)=ex, then,
dy/dx=ex, d2y/dx2 = ex; so,
ex = 1 + x/1! + x2/2! + x3/3!....
In the same way,
e-x = 1 - x/1! + x2/2! - x3/3!....
If f(x) = sin x, then,
dy/dx=cos x, d2y/dx2 = -sin x; so,
f(0) = 0;
sin x = 1 - x3/3! + x5/5!....
If f(x) = cos x, then,
dy/dx= -sin x, d2y/dx2 = -cos x; so,
f(0) = 1;
cos x = 1 - x2/2! + x4/4! - x6/6!....
Show that esin(x) = 1 + x + x2/2 -x4/8 +...
From Maclauren's Series,
sin(x) = x - x3/3!...
esin(x) = ex - x3
ex x e-x3/6
(ex = 1 + x/1! + x2/2! + x3/3!...)(e-x3/6 = 1 + x-6/3!...)
So, esin(x) = 1 + x + x2/2 -x4/8 +...
Now work out the following:
- Show Taylor Series to prove that 1/(1 + x) = 1 -x + x2 - x3 + x4 + ...
- Find an expression for tan x, using Taylor Series.
- Show that ln(1 + x) = x - x2/2 + x3/3 - x4/4...
- Use Taylor Series to find esin x.
- Use Taylor Series to find ecos x.