Mathematics I (Calculus) - Old Questions

3. (a) Find the Maclaurin series for e^x and prove that it represents e^x for all x.

Now,

Let d is any positive number with  then 

So, by Taylor’s inequality

 for 

Since  is a finite value. So

i.e.

 for all values of x.

 This implies that series converges to e^x for every value of x.