Скачать или смотреть Real Analysis Exam 3 Review Problems and Solutions

Real Analysis topics: 1) Riemann integration, 2) Fundamental Theorem of Calculus, 3) Convergence of numerical series (geometric series, ratio test), 4) Pointwise and uniform convergence of sequences of functions, 5) Weierstrass M-test (uniform convergence of series of functions). https://amzn.to/3GgFjcc ("Real Analysis", by Russell Gordon)

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(0:00) Definition of series convergence (related to sequence of partial sums)
(2:36) Absolute convergence definition
(4:44) Definition of pointwise convergence of a sequence of functions
(7:22) Definition of uniform convergence of a sequence of functions on an interval
(10:59) Ratio Test (involving limit superior and limit inferior: limsup and liminf)
(17:02) Fundamental Theorem of Calculus
(23:17) Weierstrass M-Test
(26:36) Riemann integrability and continuity
(27:22) Alternating harmonic series
(28:48) Terms of a series and convergence (including Divergence Test)
(29:33) Sum 1/k! as k goes from 0 to infinity
(29:48) Sum a geometric series
(32:49) Apply Ratio Test to decide convergence or divergence (or no conclusion)
(34:52) Use Fundamental Theorem of Calculus (along with Chain Rule to differentiate an integral)
(38:29) Taylor series calculation using geometric series (and algebraic tricks) (Radius of convergence)
(44:27) Ratio Test & integrate a Taylor series
(52:22) Geometric series & Weierstrass M-test application (geometric series of powers of cosine squared gives cotangent)
(1:00:46) Prove Mean Value Theorem for Integrals
(1:06:18) Prove Substitution Theorem (Change of Variables for a definite integral) using the Fundamental Theorem of Calculus and the Chain Rule
(1:14:20) Prove a step function is Riemann integrable

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