What is the difference between a Taylor series and a MacLaurin series?

The Taylor series and MacLaurin series are closely related mathematical concepts used to approximate functions with infinite polynomials. Here is the detailed explanation of the difference between the two:

1. **Taylor Series**:
- The Taylor series is a representation of a function as an infinite sum of terms, calculated from the derivatives of the function at a specific point.
- The general formula for the Taylor series of a function f(x) centered at \(a\) is:
f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^2 / 2! + f'''(a)(x - a)^3 / 3! + ...
This can be written as:
f(x) = Σ [f⁽ⁿ⁾(a) / n!] (x - a)ⁿ, where the summation is from n = 0 to ∞.
- The key characteristic of the Taylor series is that it is centered around a specific point \(a\), which can be any real number.

2. **MacLaurin Series**:
- The MacLaurin series is a special case of the Taylor series, where the function is expanded around \(a = 0\) (the origin).
- The general formula for the MacLaurin series is:
f(x) = f(0) + f'(0)x + f''(0)x^2 / 2! + f'''(0)x^3 / 3! + ...
Or in summation notation:
f(x) = Σ [f⁽ⁿ⁾(0) / n!] xⁿ, where the summation is from n = 0 to ∞.
- The MacLaurin series is simply a Taylor series centered at \(a = 0\).

3. **Key Difference**:
- The Taylor series can be centered at any point \(a\), while the MacLaurin series is always centered at \(a = 0\).
- MacLaurin series is a subset of Taylor series, specifically applicable when expansions around the origin are required.

4. **Example for Clarity**:
- Taylor series for e^x around \(a = 1\):
e^x = e^1 + e^1(x - 1) + e^1(x - 1)^2 / 2! + ...
- MacLaurin series for e^x (Taylor series for e^x around \(a = 0\)):
e^x = 1 + x + x^2 / 2! + x^3 / 3! + ...

In summary, the MacLaurin series is a specific type of Taylor series centered at \(a = 0\), while the Taylor series is a more general form that can be centered at any point.