Taylor Series

Taylor Series is based on a theorem that states, any smooth function, 𝑓(𝑥), can be expressed as a polynomial in the neighborhood of a point 𝑎. Assuming all derivatives of the function exist at 𝑎, the series takes the form

f(a) = C₀

The derivative of f(x) by differentiating the individual terms,

the derivative is also a power series, then computing all of its higher derivatives

when the function is evaluated at a, the constant term is obtained for each power series.

f'(a) = 1 · C₁

f''(a) = 2 · 1 · C₂

f'''(a) = 3 · 2 · 1 · C₃

.. . . . 

f^(k)(a) = k! · C_k

Solving the equation for the k-th coefficient C_k,

If function has a power series expansion at a with radius of convergence R > 0, that is,

then C_n,

Substituting C_n value in the formula, 

The above series is called Taylor series of the function f at a.

When  a = 0, the series becomes

which is called  Maclaurin series.