Practical Session 5: Power Series and Interval of Convergence

This practical session develops students' skills in working with power series and determining their intervals of convergence. The material is intended for students studying mathematical analysis who are familiar with series convergence tests.

The session covers finding radii and intervals of convergence through systematic problem-solving. Students apply the Ratio Test and Root Test to various series, including those centered at nonzero points, series with factorials, and series containing only even powers. Special attention is given to endpoint analysis, where convergence must be checked separately.

Problems progress from basic series to more challenging examples requiring careful simplification. The session also explores applications of differentiation and integration of power series, demonstrating how to derive new series representations from known ones. Each problem includes complete step-by-step solutions. Homework problems and a summary of key concepts are provided.