Advanced Calculus fi..

Advanced Calculus, Fifth EditionPROBLEMS1. Determine convergence or divergence of the following improper integrals:em+dxa) 7 dx b, JI [IO~X)~C) jp" 9d) JOo0 tke-st dt, k > -1e) sinx2 dx (Hint: Let u = xZ.)2. Prove that if f (x) = sin 2nx, then the seriesconverges, but Jr f (x) dx diverges.3. Prove the ratio test for integrals: If f (x) is continuous for a 5 x < oo andthen ST f (x) dx is absolutely convergent.4. Apply the ratio test of Problem 3 to prove convergence of the integrals:a) JF 5 dx b) 17 $ dx5. Prove the root test for integrals: If f (x) is continuous for a 5 x < oo andlim 1 f(x)l'lx = k < 1,X'"then S,bO f (x) dx is absolutely convergent.6. Apply the root test of Problem 5 to prove convergence of the integrals:00 dxa) J,OO e-l2 dxb, S2 (10gx)~7. Prove that the following integrals are uniformly convergent for 0 5 x 5 1:dta) iy (x2 + 12);8. a) Prove that for every choice of xl > 0 the integralb) SpO X* dtpe-xtz dt, n > 0,is uniformly convergent for x 2 XI.b) Use the known result that1I" 2(Problem 1 following Section 4.8) to prove thate-X2 dx = - fiC) Use the results of parts (a) and (b) and Theorem 58 to show that for n = 1,2, . . . ,and x > 0,I"9. a) Prove that if n > 0, the integralstne-t2 cos (tx) at, pe-t2 sin . (tx) dt1"are un~formly convergent for all x .