Taylor’s and Maclaurin’s Series

Taylor’s theorem states that we can get expansion of a function f(x) at x=a
This can be written as

When we put a=0 it becomes Maclaurin’s series, and it has a lot of applications in mathematics and other areas.

If you want expansion of e^x you will require to find f(0), f’(0), f’’(0), f’’’(0), …………., fⁿ (0)……..
And fortunately we will get
f(0)= f’(0)=f’’(0)=f’’’(0)=…….=f ⁿ (0)……=1
After putting these values in the above equation we will get

The same way we can find out the expansion of sin(x), after calculating
f(0), f’(0), f’’(0), f’’’(0), …………., ⁿ (0)……..
Here f(0)=sin(0)=0
f’(x)= cos(x) and f’(0)=cos(0)=1
f’’(x)=-sin(x) and f’’(0)=-sin(0)=0
f’’’(x)=-cos(x) and f’’’(0)=-cos(0)=-1
f’’’’(x)=sin(x) and f’’’’(0)=sin(x)=0
f’’’’’(x)=cos(x) and f’’’’’(0)=1
………………………………………………..
…………………………………………………
And if we put the above in Maclaurin’s series we will get the following series.