what are the applications of taylor's series & laurent's series?

Answer 1

You probably learnt Maclaurin and Taylor series in high school math, right? Well these are basically series derived for a function f(x) using values of its derivatives at certain points and combining it with an approach much like the binomial theorem expansion. Maclaurin series are actually a type of Taylor series which consider derivatives at x=0,(where x is the variable)these are the simplest series of this form. Taylor series can be used with any number to plug into the variable without it necesarily being zero. Also these series are valid when the domain of the function is complex. And it is worth noting that these power series are ascending in degree of the variable within the set of positive integers only.
As for applications these series come in handy for proving properties of the functions they represent where the error term is negligible or does not exist. For example,
e^ix = cos x + isin x is often shown to be true with series expansions.
Also evaluating certain limits, mostly as x->0, can be done using these expansions.
And for really hard to evaluate integrals(in numerical analysis) integrating the series expansion term by term is valid and depending on the accuracy required more and more terms could be considered.
Also solving differential equations is a possibility.

As for Laurent series I stressed that Taylor series are of ascending positive degrees of the variable. In certain situations Taylor series can't be used because of this. Laurent series are a generalisation which span over the entire set of integers(-ve and +ve) for the degree. A line integral defines the coefficient here.

Laurent series primarily express complex functions and have about the same applications as listed above.

Also, it is to be noted that only infinitely differentiable functions may be expressed with these power series, making it all the more powerful in complex analysis for the study of analytic functions. These series may also appear in several variables in which case partial derivatives are to be considered.

Answer 2

Basically, Taylor's Theorem states that every function can be written as an infinite series and gives the criteria and formula to do so.

This is a powerful tool in mathematics. It allows you rewrite functions, solve equations (to a given decimal place), etc. Ever seen pi as an infinite series?

If you continue on in your mathematical career, you will encounter this many times.

Answer 3

taylor series is used to find logarithm.
about laurent's.....???????!!!!???? dont know