Analysis Review Question: Limits and Integrals

Please help me with the attached question.

Power Series Convergence

Please help me with the attached question.

Sigma-algebra

(1) Let S = { E_1, ..., E_n} be a collection of nonempty subsets of a set X such that E_1 U .. U E_n = X and the E_i are pairwise disjoint. See attached for complete questions.