Taylor series expansions can be derived using integration from "summary" of Skills in Mathematics - Integral Calculus for JEE Main and Advanced by Amit M Agarwal
To understand how Taylor series expansions can be derived using integration, we need to start by recalling the definition of a Taylor series. A Taylor series is an infinite sum of terms that represent the values of a function at a particular point in terms of its derivatives at that point. The general form of a Taylor series for a function f(x) centered at a point a is given by:f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ... To derive this series using integration, we can start by writing the nth derivative of the function at the point a in terms of an integral. This can be done by using integration by parts or other integration techniques to express the nth derivative in a form that involves the function itself and its derivatives up to the (n-1)th order. By integrating the nth derivative term by term, we can obtain an expression for the nth term in the Taylor series expansion. This is because integration is the reverse operation of differentiation, and by integrating each term of the nth derivative, we can retrieve the original function and its derivatives up to the (n-1)th order. By continuing this process for all values of n, we can derive the complete Taylor series expansion of the function centered at the point a. This series will be an infinite sum of terms that represent the function and its derivatives at the point a, providing a way to approximate the function using a polynomial expression. In summary, the process of deriving Taylor series expansions using integration involves expressing the nth derivative of a function in terms of an integral and then integrating each term of the nth derivative to obtain the corresponding term in the Taylor series expansion. This allows us to represent a function as an infinite sum of terms involving its derivatives, providing a powerful tool for approximating functions and analyzing their behavior at a particular point.
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