Taylor Series Theory and Examples Short Cut Trick Method GATE

Taylor Series ( IIT JEE and GATE Mathematics Short Cut Method or Short Trick )

If a function f(x) is continuous and infinitely differentiable, then its Taylor series about x = a is given by

f(x) = f(a) + f’(a) (x-a) + (1/2!)f’’(a)(x-a)^2+…

Note that Taylor series contain only nonnegative ( zero and positive ) powers of (x-a).

If Taylor series of a function exists, then it is unique. We can use any valid mathematical manipulation to find the Taylor series of the function.

Example : The third term in the Taylor’s series expansion of e^x about x = a would be

Example : The Taylor series expansion of sin x about x = / 6 is given by

Example : Which of the following functions would have only odd powers of x in its Taylor series expansion about x = 0

Example : The Taylor series expansion of sin x / ( x -  at x =  is given by

Example : In the Taylor series expansion of e^x + sin x about the point x = , the coefficient of ( x - ^2 is

Example : For the function e^(-x), the linear expansion around x = 2 is

Example : for |x| very very smaller than one 1, coth (x) can be approximated as


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