6 Taylor Series

6.4 Examples from our Basic Set of Functions

6.4.3 Exponential Functions

An especially easy calculation is the series of the natural exponential function:

math formula

since math formula  with math formula .

In particular, for math formula we get math formula , the series through which we have defined the  number e .

math formula
Figure 6.4: The Taylor series for the exponential function

Insert: The limit from Exercise 4.14d


The inverse function of the natural exponential function, the natural logarithm, can not be expanded around math formula, since math formula. There is however a series for
math formula


Exercise 6.4:

Prove this Taylor series for math formula. Solution



In the following table we have put together for you the  first two terms of the Taylor series for some common functions, to make memorization easier.

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