2. One example of an integral that is difficult to solve without using a Taylor Series is the function e . The integral of the function would be fore " da . Apparently this function is known as the error function or er f (a). It's challenging to solve as an integral, but it can be evaluated with the Taylor Series expansion centered at x = 0 which is e ~ ~ 1 -x2+ - 2 + . . . which can be integrated term by term. The approximation of the is therefore 10 42 + 216 - . . . and summing up more and more terms, the value of the expression becomes arbitrarily close to the integral. 3. An application of a Taylor Series in economics and finance is a Taylor Series that can be used to approximate the movement in value of a derivatives contract. AP = f' (S)AS + 2 f\\doubleprime ( S) (AS) ? + ... In the equation above, the price of the asset underlying the contract is S, the first derivative is called the delta and the third derivative is called the gamma. Traders and investors leverage this information to make asset management decisions