Taylor series expanded around the point \( x=0 \)

A type of Taylor series that expresses a function as an infinite sum of terms derived from the function's derivatives at zero

Named after Colin Maclaurin

A series named after the Scottish mathematician Colin Maclaurin, used for approximating complex functions with polynomial terms

General form of Maclaurin series

The general form of a Maclaurin series for a function \( f \), which is infinitely differentiable at \( x=0 \), is \( M_f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n + \cdots \)

Expanded form

The expanded form lists each term individually

Sigma notation

The sigma notation condenses the series into a more compact mathematical expression

Construction of Maclaurin series

To construct a Maclaurin series for a specific function, one must calculate the derivatives of the function, evaluate them at \( x=0 \), and then assemble the series using these values

Natural logarithm function

The Maclaurin series for the natural logarithm function \( f(x) = \ln(1+x) \) is \( M_f(x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \)

Cosine function

The Maclaurin series for the cosine function \( f(x) = \cos(x) \) is \( M_f(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \)

Validity of Maclaurin series

The validity of the Maclaurin series is established through a proof similar to that of the Taylor series, relying on the concept that the series converges to the function within its interval of convergence

Power series expansion

Maclaurin series can be used to find expansions for more intricate functions, such as \( f(x) = x^2e^x \)

Hyperbolic cosine function

The Maclaurin series for the hyperbolic cosine function \( f(x) = \cosh(x) \) can be derived by either calculating the derivatives or by employing the exponential definition of hyperbolic cosine

Interval of convergence

The Ratio Test is employed to ascertain the interval of convergence for a series, involving calculating the limit of the absolute value of the ratio of successive terms and ensuring it is less than one