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.

 

 

Recommended Reading

 

Maths is challenging; so is finding the right book. K A Stroud, in this book, cleverly managed to make all the major topics crystal clear with plenty of examples; popularity of the book speak for itself - 7th edition in print.

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