Assume that the Maclaurin series converges to the function. Show that \(1-\cos x=x^{2} / 2\) with an error less than \(0.003\) for \(|x|<\frac{1}{2}\).

The cosine function, denoted as \(\text{cos} \thinspace x\), is one of the fundamental trigonometric functions. It describes the x-coordinate of a point on the unit circle corresponding to an angle \(x\) measured from the positive x-axis. It has several key properties: it is an even function, periodic with a period of \(2\thinspace \text{π}\), and its values range between -1 and 1.
In calculus, the cosine function can be approximated using a polynomial expression derived from its Maclaurin series. This helps in simplifying computations, especially for small values of \(x\). The Maclaurin series takes advantage of the function's derivatives at zero. For the cosine function, this series is:
\(\text{cos} \thinspace x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \thinspace \text{...}\)
This infinite series allows us to approximate \( \text{cos} \thinspace x\) with a desired accuracy by truncating the series after a few terms.

When approximating functions using series expansions, it is crucial to understand the concept of error bound. The error bound gives us a measure of how far off our approximation might be from the actual value. For the cosine function, truncating its Maclaurin series means omitting higher-order terms, and the error bound captures the impact of these omitted terms.
In our example, the series is approximated as
\(\text{cos} \thinspace x \thickapprox 1 - \frac{x^2}{2}\).
The next term in the series that we exclude is \(-\frac{x^4}{24}\).
For \(|x| < \frac{1}{2}\), the error term is bounded by \(\frac{x^4}{24}\).
Calculating the maximum error:
\(\frac{(\frac{1}{2})^4}{24} = \frac{1}{384} \thickapprox 0.0026\)
This ensures that the approximation error is within the specified tolerance of 0.003. Thus, the high accuracy of our approximation \( \text{cos} \thinspace x \thickapprox 1 - \frac{x^2}{2}\) is confirmed for small values of \(x\).

Series expansion is a powerful tool in calculus used to represent complex functions as infinite sums of simpler terms. By breaking down functions into series, we can study and approximate them more easily. The Maclaurin series is a specific type of series expansion centered at \( x = 0\).
For a function \( f \thinspace (x)\), the Maclaurin series is given by:
\ f(x) = f(0) + f'(0) \frac{x}{1!} + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 + ...\
This series provides an accurate approximation by including terms up to the desired degree based on the derivatives of the function evaluated at zero.
The cosine function's Maclaurin series expansion:
\( \text{cos} \thinspace x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...\)
demonstrates how a seemingly complex function can be expressed as a sum of more straightforward polynomial terms. This expansion is particularly invaluable in practical applications, as it allows for simplifying otherwise complex computations.

In mathematics, the convergence of a series is an important concept that ensures the series sums to a finite value as more terms are added. A series is said to converge if its sequence of partial sums approaches a specific value.
The Maclaurin series for \( \text{cos} \thinspace x\) converges for all real numbers. This means that as you include more terms in the series, the approximation of \( \text{cos} \thinspace x\) gets closer to its true value. The convergence property is crucial because it justifies using a finite number of terms to achieve desired accuracy.
For example, in our case:
\(1 - \text{cos} \thinspace x \thickapprox \frac{x^2}{2} - \frac{x^4}{24} \thickapprox \frac{x^2}{2} \)
for small values of \( x \), aligns perfectly because the series converges rapidly. Ensuring convergence guarantees reliable and precise approximations, which is foundational in fields requiring numerical analysis and computational methods.