Upper and Lower Sums - Round Two (a) Consider the following partition sequences for the interval [0, 1]: Zn = (x0, x1, ..., xn), with xk = ½-½ Zn = (0, T1, T2, ..., Tn), with Ãk = 3 Determine the fineness measures | Z | and |Z❘ depending on n, as well as their limits for n → ∞, and justify why Zn is unsuitable for determining the integral. (b) Using the partition Z from (a), determine the integral So √x dx. Then slightly modify the partition and use it to determine the integral over the interval [0,6] for any bЄ R. Hint: You may use the summation formulas in this problem without proof: ΣΕ 1 k² = n(n+1)(2n + 1) and 1 k = n(n + 1).

Real Analysis For each n in the positive integers, let Pn=<xn,yn> be a point in R2. Show that {Pn} from n=1 to infinity converges to P=<x,y> in R2 if and only if {xn} from n=1 to infinity and {yn} from n=1 to infinity converge in R1 to x and y respectively. Please be thorough in your answer, explaining each portion to me.  Thank you.

Numerical Analysis

T or F , prove your answer Suppose (an) is Cauchy and that for every n ∈ N, the interval (−1/n, 1/n)contains infinitely many terms of (an). Then (an) converges to 0.