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 op...
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