Question

What are some real life applications of using taylor series expansions?

Answer 1

They can be used by computer programs to compute values of sine, cosine, and tangent or logarithm functions
Eg. sine can be computed (in radians) using:

`sin x = x-x^3/(3!)+x^5/(5!) - ...`

Similarly, they can be used to provide approximations to square roots, cube roots etc. Eg

`sqrt(1+x)=1+1/2x-1/8 x^2-1/16x^3 -5/128x^4 ...`
`:. sqrt(2)=1+1/2-1/8 -1/16 -5/128 ...`

Also, they can be used to approximates fractions:

`1/(1+x) = 1-x+x^2-x^3+ ...`

They can be used to evaluate limits:
Eg:

`\ \ \ \ \ \ sin x = x-x^3/(3!)+x^5/(5!) - ...`
`:. sin x/x = 1-x^2/(3!)+x^4/(5!) - ...`
`:. lim_(x rarr 0)sin x/x = 1`, a well known limit

They can be used to provide linear or quadratic approximations to functions

And they can be used to provide power series expressions for functions that cannot be integrated. Eg `e^(x^2)`

`e^x = 1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + ...`
`:. e^(x^2) = 1 + x^2 + x^4/(2!) + x^6/(3!) + x^8/(4!) + ...`
`:. inte^(x^2)dx = int{1 + x^2 + x^4/(2!) + x^6/(3!) + x^8/(4!) + ... }dx`
`:. inte^(x^2)dx = x + x^3/3 + 1/5x^5/(2!) + 1/7x^7/(3!) + 1/9x^9/(4!) + ...`