Find out the Taylor Series for f (x) = e^x about x = 0.

Solution

In fact this is one of the easier Taylor Series that we'll be asked to calculate.  To find out the Taylor Series for a function we will need to find out a general formula for f⁽ⁿ⁾(a).  This is one of the some functions in which this is simple to do right from the start. 

To obtain a formula for f⁽ⁿ⁾(0) all we should require to do is recognize that,

f(n) (x) = e^x                             n = 0,1,2,3,....

and so

f(n) (0) = e⁰ = 1                       n = 0,1,2,3,...

Hence, the taylor series for f (x) = e^x about x=0 is,