Lecture 6: Power Series Expansions and Applications

This lecture explores the representation of functions as power series, focusing on Taylor and Maclaurin series and their practical applications. The material is intended for students studying mathematical analysis who are familiar with power series convergence.

The lecture begins with Taylor's Theorem, providing the foundation for approximating functions by polynomials. Maclaurin series for elementary functions are derived, including exponential, trigonometric, logarithmic, and binomial series. A summary table presents these essential series and their intervals of convergence.

The second part demonstrates applications of power series: approximating definite integrals without elementary antiderivatives, solving differential equations, evaluating limits, and summing infinite series. Error estimation using Taylor's Inequality is also covered. The lecture concludes with additional examples and a summary of key concepts.