Consider the following series:

The above series is known as power Series in x.

The power series converges for some values of x and diverges for others.

If the above power series is equal to some f(x) and then for all values of x within the interval of convergence, the function f satisfies the following 3 properties.

a) f is continuous b) f is differentiable c)f is integrable

MacLaurin Series:

This Series can be derived from the first 2 properties of the above Power Series as follows.

If f(x) = a₀ + a₁x +a₂x² +a₃x³ + ...

Then, f^'(x) = a₁ + 2a₂x + 3a₃ x² +4a₄x³+ ...

f^''(x) = 2a₂ + 3.2a₃x + 4.3a₄ x² + 5.4a₅x³+ ...

f^'''(x) = 3.2a₃ + 4.3.2a₄x + 5.4.3a₅ x² + ...

If x =0, then we have f(0) = a₀, f^'(0) = a₁, f^''(0) = 2a₂ , f^'''(0) = 3.2a₃ =6a₃

In general, we can write f⁽ⁿ⁾(0) = n! a_n

and a_n = f⁽ⁿ⁾(0)/n!

Substituting these a_n values in the equation for f(x), we get Maclaurin Series as follows.

f(x) = f(0) + f^'(0)x + {f^''(0)}x²/2! +{f^'''(0)}x³/3! + ... +{fⁿ(0)}xⁿ/n! + ...