Lista 1 - Cálculo 4

Lista de exercicios (extraida do livro de Boyce e Di Prima, secdo 5.2, pag. 195) Problems In each of Problems 1 through 6, determine the radius of convergence ge . : 15. Lety = So a,x". of the given power series. ~, Co 1. SO(«-3)" a. Compute y’ and y” and write out the first four terms of each n=0 series, as well as the coefficient of x” in the general term. © x b. Show that if y” = y, then the coefficients ag and a, are 2. u ant arbitrary, and determine az and as in terms of ag and ay. n=C in c. Show that a,.. = ———————.,n = 0, 1, 2,3, ..... xo y2n P+ 2Dn+ Sr | n=0 "" In each of Problems 16 and 17, verify the given equation. Co Cc Co 4. So 2"x" 16. So an(x — 1)" = So a,_\(x — 1)" n=0 n=0 n=1 (X= X90)" . kK ke > k 5. S> ———— V7. So agg x® +O agx®*! = ay + O(a + ay) x n=l n k=0 k=0 k=1 9 (=1)"n?2(x +2)" In each of Problems 18 through 22, rewrite the given expression as a 6. u 3n single power series whose generic term involves x”. Cc In each of Problems 7 through 13, determine the Taylor series about 18. > n(n — l)a,x"~? the point xo for the given function. Also determine the radius of n=2 convergence of the series. oo oo 7 si 19. x SO Nayx"! + > apx* - sinx, x=0 n=l k=0 8. e*, =0 °c °c c *o 20. SS mim — L)dnx”~? + x > kayx*—! 9. x, x= 1 m=2 k=l Cc Cc 10. x2, x =-l 21. So nayx™! +x 0 ayx" ll. Inx, x=1 nol n=0 Cc Co 1 22. x So n(n = Vanx"-? + SO nx” 12. T-x _ x > xy = 0 n= n=O 1 23. Determine the a, so that the equation 13. T-x xy = 2 —x co oo 00 S> nanx" 1 +2 S> ax" =0 14. Let y = So nx". nal nad n=0 a. Compute y’ and write out the first four terms of the series is satisfied. Try to identify the function represented by the series ° . Co b. Compute y” and write out the first four terms of the series. So dy x”. n=0