Maclaurin Series for Cosine of Double Angle

Using the Maclaurin series for $\cos(x)$, rewrite the series to accommodate $\cos(2x)$, and simplify the expression as necessary.

The exploration of Taylor and Maclaurin series is an essential topic in calculus, as these series provide a means to approximate complex functions with power series. The Maclaurin series is a special case of the Taylor series centered at zero, and it is used to approximate functions like sine, cosine, and exponential functions using an infinite sum of terms. The cosine function, in particular, is often expanded into a series using its even symmetry and periodic properties, which make it a useful function to study in mathematical analysis.

In this problem, we delve into the Maclaurin series representation for the function cosine of 2x. The Maclaurin series for cos(x) is given by the sum of successive even-powered terms of x, each multiplied by alternating signs and divided by the factorial of the exponent. When extending this series to accommodate cos(2x), one must adeptly substitute and simplify terms in the series to account for the doubled angle. This task highlights not only the process of substituting variables within series expansions but also the need for careful algebraic manipulation to simplify expressions to their most compact form.

Understanding how to manipulate and simplify series expansions like these plays a fundamental role in solving more complex problems involving differential equations and in approximating solutions to integrals. By developing proficiency in handling series expansions, students gain valuable skills applicable to various fields such as physics, engineering, and computer science, where function approximations are pivotal.