Taylor Series

Let be infinitely many times differentiable on and .

If for , then Taylor series of at is given by:

Note

Usually Taylor series expansion is done with . This is a special case of Taylor series, and called the Maclaurin series.

Procedure

Suppose a function is given and its Taylor series is required.

• Differentiate repeatedly and find a general solution for -th derivative
• Construct Taylor polynomial
• Use root test or ratio test to find the range of convergence of the Taylor polynomial
• Consider the endpoints of range of convergence to check if the Taylor polynomial converges
• Construct the Taylor remainder
• Find for which values of , the remainder converges to

Examples

e^x

Range of convergence is .

ln (1+x)

Range of convergence is .

sin x

Range of convergence is .

cos x

Range of convergence is .