The integral \(\int_0^x \frac{1 - \cos(t)}{t} dt\) can be expressed as an infinite series using the Taylor series expansion for cosine. Specifically, the series expansion is given by \(\cos(t) = \sum_{n=0}^{\infty} \frac{(-1)^n t^{2n}}{(2n)!}\). By substituting this series into the integral, the term '1' in the numerator can be canceled, simplifying the integration process. This approach allows for the evaluation of the integral in terms of a series representation.
PREREQUISITESStudents and educators in mathematics, particularly those focusing on calculus and series analysis, as well as anyone seeking to deepen their understanding of integral representations and series expansions.