4n – 3 1 1 + + 22 1 1 Yn = 12 ... 2n 32 n2 In this exercise we will show that they are convergent. (a) Show that (xn) is eventually decreasing and bounded below. By eventually decreasing it is meant that Xn+1 < Xn, for large enough n E N. (b) Show that (yn) is increasing and bounded above.

Numerical Analysis

k Σ (k² + 2)3/2 Use the integral test to determine if converges. k=1 8.

∞ Suppose ak > 0 for all k, and ∑ ak converges.                                                   k=1 Answer the following questions and explain your reasoning.               ∞        a. Must ∑−3an always converge?               n=1               ∞b. Must ∑ 1/an always converge?              n=1              ∞ c. Must ∑(an )^3 always converge?              n=1             ∞ d. Must ∑ (an)^1/3 always converge?               n=1

n k+1 Suppose (ar) is monotone decreasing and ap → 0. Set sn = (-1)*+'ak. Show that the subsequence k=1 (82n-1) converges.

Proof the following theorem.

Vn+1 E(-1)" n+1 n=1 O diverges C) converges

SOLVE STEP BY STEP IN DIGITAL FORMAT PLEASE Through the point P(-2,-7) tangents are drawn to the ellipse 2x² + y² + 2x-3y - 2 = 0; determine the equations of the tangents and the points of tangency +