Write the Maclaurin series expansion of the following functions:

• sin x

The Maclaurin series is a special case of the Taylor series, centered at zero. For the sine function, the Maclaurin series expansion can be derived from its derivatives. Here’s how it looks:

Maclaurin Series for sin x

The series expansion for sin x is given by:

• sin x = x - (x³/3!) + (x⁵/5!) - (x⁷/7!) + ...

General Formula

This can be expressed in summation notation as:

sin x = Σ (-1)^n * (x^(2n + 1)) / (2n + 1)!, where n = 0, 1, 2, ...

Understanding the Terms

Each term in the series alternates in sign and involves odd powers of x, divided by the factorial of the corresponding odd number. This pattern continues infinitely, allowing for an approximation of sin x for values of x near zero.

Applications

The Maclaurin series for sin x is particularly useful in calculus and physics, especially when dealing with small angle approximations or in numerical methods.