### 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)=ex, then,
dy/dx=ex,    d2y/dx2 = ex; so,
f(0)=1;
f'(0)=1;
f"(0)=1;

So,

ex = 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 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:

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 esin x.
5. Use Taylor Series to find ecos x.

### Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

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