What are Taylor series used for?

If you wanted to program a function into a computer, it knows how to calculate polynomials but not any function that you put into it. For example, it won't know what to calculate for sin(x), but it would know what to do if I told it to calculate:x - x³/3! + x⁵/5! - x⁷/7! ...Now, there's a problem in the fact that the computer can't calculate an infinitely long polynomial, so you have to stop at a certain point and use a 'truncated Taylor series'.
Outside of the computing side of it, another use of Taylor series is for differential equations. Not all differential equations are solvable by nice analytic methods, so we have to approximate the answer. If we have the correct initial values for a differential equation, we can continue to differentiate it for the coefficients to the Taylor series. When we've done this for a suitable amount of coefficients, we again use a truncated Taylor series for an approximate answer to the differential equation.