Ideas for exam 1: Chapter 22 Definition of sequence and convergence of a sequence. Proving the limit of a sequence using epsilon Theorem 1 and Theorem 2 and their proofs. Definition of a Cauchy sequence Theorem 3 and the Bolzano-Weierstrass Theorems. Be able to find limits of sequences and prove it's the limit Be able to evaluate limits that are Riemann sums Proving basic limit theorems. Chapter 23 The vanishing condition Geometric Series and Partial Fraction series Comparison test and its proof Limit comparison test Ratio test and its proof Integral test Leibniz's theorem Important series like sin(x) Be able to determine whether a series converges or diverges and justify your answer Evaluate infinite products. Some sample problems: p. 453 1(vi) p. 453 2(i) and prove it's the limit Find lim (as n-> infinity) (1 + 1/2n)^n p. 455 9(i),(iii),(v) Prove the sequence x_1 = sqrt(2); x_{n+1} = sqrt(2x_n) is monotone and bounded and find its limit. p. 483 1(i) (v) (ix) (xv) (xvii) (xix) p. 483 4. p. 483 26 Does SIGMA(from n=3 to infinity) (1/n) exp(-sqrt(log(n))) converge or diverge, justify Does SIGMA(from n=1 to infinity) (n^{1/n} - 1) converge or diverge, justify Give an example of a convergent series that is not absolutely convergent. Explain why the example is convergent but not absolutely convergent. Prove: If {a_n} and {b_n} are convergent sequences of real numbers and a_n <= b_n for all n sufficiently large, then lim (n-> infinity) a_n <= lim (n-> infinity) b_n. Evaluate SIGMA (from n=3 to infinity) 1 / (4n^2 - 1) Chapter 24 What does f_n(x) converge to? Does it converge uniformly (probs 1 and 2) Write an infinite series as a function and find its interval of convergence (problem 4). Find an infinite sum by recognizing it as a series of a "nice" function. (problem 5) Prove a series converges uniformly on an interval (Weierstrass-M test)