Definition of a sequence. Theorems on limits of sequences. Bounded and monotonic sequences. Cauchy’s convergence criterion. Series of non-negative terms. Comparison tests. Cauchy’s integral tests. Ratio tests. Cauchy’s root test. Raabe’s test logarithmic test. De’morgan’s and Bertrand’s tests. Kummer’s test, Cauchy Condensation test, Gauss test, alternative series. Leibnitz’s test, absolute and conditional convergence.
Partitions, Upper and lower sums. Upper and lower integrals, Riemann integrability. Conditions of existence of Riemann integrability of continuous functions and of monotone functions. Algebra of integrable functions. Improper integrals and statements of their conditions of existence. Test of the convergence of improper integrals. Beta and Gamma functions.