This excerpt is taken from a Project Gutenberg publication of A Course of Pure Mathematics by G.H. Hardy. The text covers fundamental concepts in mathematics, including the definition and properties of real numbers, functions, limits, continuity, and derivatives. It explores the relationship between geometry and algebra, particularly how geometrical concepts can be expressed using mathematical equations and how the theory of functions can be applied to understand curves. The text also covers the theory of infinite series and integrals, exploring their convergence and divergence and how they can be used to solve problems in calculus.

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