A Taylor Series is a mathematical representation of a function using an infinite sum of terms, each calculated based on the function's derivatives at a specific point.
To find the Taylor Series for a function, you need to calculate its derivatives at a specific point and then use those values to create the series. The formula for a Taylor Series is: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ... + f^n(a)(x-a)^n/n!
The Taylor Series for y(x)=sin^2x is: y(x) = x^2 - x^4/3! + x^6/5! - x^8/7! + ... + (-1)^n * x^(2n)/(2n+1)!
The Taylor Series allows us to approximate complex functions with simpler ones, making it easier to solve mathematical problems. It also helps in understanding the behavior of a function around a specific point.
The Taylor Series has various applications in mathematics, physics, and engineering. It is used to approximate functions in numerical analysis, solve differential equations, and in signal processing. It is also used in calculus to understand the behavior of functions and in statistics to analyze data.