Taylor Series

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:

E.g.1

If f(x)=e^x, then,
dy/dx=e^x,    d²y/dx² = e^x; so,
f(0)=1;
f'(0)=1;
f"(0)=1;

So,

e^x = 1 + x/1! + x²/2! + x³/3!....

In the same way,

e^-x = 1 - x/1! + x²/2! - x³/3!....

E.g.2

If f(x) = sin x, then,
dy/dx=cos x,    d²y/dx² = -sin x; so,
f(0) = 0;
f'(0)=1;
f"(0)=0;

So,

sin x = 1 - x³/3! + x⁵/5!....

E.g.3

If f(x) = cos x, then,
dy/dx= -sin x,    d²y/dx² = -cos x; so,
f(0) = 1;
f'(0)=0;
f"(0)=-1;

cos x = 1 - x²/2! + x⁴/4! - x⁶/6!....

E.g.4

Show that e^sin(x) = 1 + x + x²/2 -x⁴/8 +...

From Maclauren's Series,
sin(x) = x - x³/3!...
e^sin(x) = e^{x - x3}
e^x x e^-x3/6
(e^x = 1 + x/1! + x²/2! + x³/3!...)(e^-x3/6 = 1 + x^-6/3!...)
So, e^sin(x) = 1 + x + x²/2 -x⁴/8 +...

Now work out the following:

1. Show Taylor Series to prove that 1/(1 + x) = 1 -x + x2 - x3 + x4 + ...
2. Find an expression for tan x, using Taylor Series.
3. Show that ln(1 + x) = x - x²/2 + x³/3 - x⁴/4...
4. Use Taylor Series to find e^{sin x.}
5. Use Taylor Series to find e^{cos x.}