Maclaurin series

• Maclaurin Series is a specific type of Taylor Series that is expanded around the point x = 0.

• The general form for a Maclaurin series is f(x) = f(0) + f’(0)x + f’‘(0)x^2/2! + f’’‘(0)*x^3/3! + …

• In the above formula, each f^n(0) represents the nth derivative of the function evaluated at x = 0.

• This series can be used to approximate functions near x = 0, offering close estimates when using a few terms and increasingly precise results as more terms are used.

• It’s essential to identify when a function can be represented by a known Maclaurin series. For example, e^x, sin(x), cos(x), and 1/(1-x) all have standard Maclaurin expansions.

• For the function e^x, the Maclaurin series is e^x = 1 + x + x^2/2! + x^3/3! + …

• For sin(x), the Maclaurin series is sin(x) = x - x^3/3! + x^5/5! - x^7/7! + …

• For cos(x), the Maclaurin series is cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + …

• For 1/(1-x), the Maclaurin series is 1/(1-x) = 1 + x + x^2 + x^3 + …

• Be aware of the radius of convergence, which is the range of x-values where the series is valid. For example, the radius of convergence for the series of 1/(1-x) is

  x

  < 1.

• Also be aware of the remainder term, Rn, which represents the error between the approximation of the function and the actual function value.

• To directly apply the Maclaurin series to problem-solving, you may need to manipulate the given function so that it resembles a standard series.

• It’s crucial to practice applying these series to different types of functions with varying levels of complexity to become comfortable with their uses.