Sequences and Series of Real Numbers: Foundational Real Analysis

Understanding how infinite lists of numbers behave is crucial for higher mathematics, calculus, and advanced data fields. Many learners struggle with the rigorous proofs and convergence tests required to master real analysis. This text-only course guides you from the absolute basics of sequences to the mechanics of infinite series. You will read clear explanations, study step-by-step mathematical proofs, and build the analytical skills needed to tackle complex mathematical problems with confidence. What you'll learn: 1. Understand the fundamental definitions of bounded, monotonic, and convergent sequences. 2. Apply key convergence tests, including the Ratio, Root, Comparison, and Integral tests, to infinite series. 3. Calculate the limits of sequences using rigorous mathematical definitions and theorems. 4. Distinguish between absolute and conditional convergence in alternating series. 5. Practice solving classical real analysis problems through structured written exercises. The course starts with basic definitions of real numbers and sequences, ensuring you have a solid foundation before moving on to limits. You will then progress systematically through limit theorems, convergence criteria, and the behavior of infinite series. Designed for undergraduate students, exam aspirants, and mathematics enthusiasts, this course requires no prior advanced calculus. Start reading today to demystify the mathematics of infinite sequences and series.