Advanced Calculus fi..

Chapter 6 Infinite Series 447d 2 = ~ Fxx(x, y)dx2 + 2Fxy(x, y)dx dy +'F,,(x, y)dy2 one obtains-rd2~(x(u, v), y(u, v)) = Fxx(x(u, v), y(u, v))at each point (ul, vl) at which x = xl, y = yl and Fx(xl, yl) = 0, Fy(xl, yl) = 0.(See Problem 6 below.)The invariance of d2F at a critical point explains its importance for maxima andminima as in the preceding paragraph. The concept also occurs in discussion of thesecondfindamental form on a surface (Problem 23(d) following Section 3.11). Seealso Sections 5.19 and 5.20.PROBLEMS1. Expand in power series, stating the region of convergence:a) e ~2-~2 b) sin(xy)1 1c) I-n-y d) 1-x-y-z2. Prove Eq. (6.69) by induction.3. Prove that if a power series.converges at (xo, yo), then it converges at every point (Axo, hyo), for 1 Al c 1.4. Evaluate J; 1; sin (xy) dx dy with the aid of power series.5. As in Problem 9 following Section 6.18, use Taylor series or the remainder formula toanalyze the following difference approximations to partial derivative expressions:a) g,(x,h) = f (x+h,y+h)+f(x+h,y-h) + f(x-h,y+h)+f(x-h.y-h)-4f(x3y)h2h2as approximation to v2f at (x, y);b) g2(x,h)= f(~+h.~+h)-f(x+h,y-h)-f(x-h,y+h)+f(x-h,~-h)as approximation to a2f/ax ay at (x, y).4i6. Prove the validity of Eq. (6.71) at (ul, vl) under the hypotheses stated. [Hint: Use therules of Section 2.16.14h2In Section 4.1, improper integrals are defined. In this section we show that there is aclose relationship between improper integrals and infinite series. The relationship issuggested by the integral test of Section 6.6. It leads us here to some valuable testsfor convergence of improper integrals. Attention will be confined to integrals froma to oo, but the methods are applicable to other forms of improper integrals.Let us consider first, as an example, the integral