Limits, Series, and Calculus

Establish equality between infinite series or products using finite truncations, absolute convergence and explicit limiting arguments. Compute Fourier coefficients and use a justified convergence theorem to evaluate an exact series sum, including endpoint behavior. Derive a limiting proportion of integer structures by comparing finite counts and controlling truncation or boundary errors. Determine the limit of a recursively defined numerical sequence and prove convergence using monotonicity, bounds, contraction or an equivalent exact argument. Estimate a definite or improper integral and prove an explicit relative-error guarantee by valid comparison, transformation or remainder bounds.