This book presents counterexamples to false statements typically found within the study of mathematical analysis, real analysis, and calculus, all of which are related to infinite sequences, series of functions, and functions and integrals depending on a parameter. In addition, the book aims to help readers master the presented concepts and theorems, which are traditionally challenging and are sources of misunderstanding and confusion. The authors provide an overview of important concepts and theorems on different types of the convergence of infinite sequences and series by employing counterexamples, and coverage was restricted to the main definitions and theorems in order to explore different versions (wrong and correct) of the fundamental concepts. The book is divided in six chapters: the first chapter contains introductory material such as comments on notations, presentation form, and background theory; the second chapter considers conditions of uniform convergence; the third chapter deals with properties of limit functions such as boundedness, existence of the limit, and continuity; the fourth chapter analyzes the conditions of differentiability and integrability of the limit functions; and the two final chapters consider the properties of integrals (proper and improper) depending on a parameter. It is assumed that readers have some knowledge of calculus. Topical coverage includes: Introduction; Conditions of Uniform Convergence; Properties of the Limit Functions: Boundedness, Limits, and Continuity; Properties of the Limit Function: Differentiability and Integrability; Integrals Depending on a Parameter; and Improper Integrals Depending on a Parameter.
