Question

(1 point) Match each of the following with the correct statement. A. The series is absolutely convergent. C. The series converges, but is not absolutely convergent. D. The series diverges. (2 + n)5" 1. (-1)n+1 (n2)521 2. Ž -on)" n=1 nin n=1 3. (-1)"nIn(n + 2) n=1 4. (n + 6)" 6" (n + 6)" n=1 5. no 612 72 6. Σ n? 10n4 8 n=1

Answer 1

h=1 [1] Ž (-1)"H (2+2) 5h (m?) (52") det an= (-1) hh (2+2)5" h2, 52n = ) land (-ighth (2+2)5h h2 52n 15"] (241) 5" 12.52n By Ratio Tent- anti n²52n clim lantil clim (n+3) 5" (n+1)2 52(n+1) (2+2)5h no lanl his n+3 25ht 529 clim nts ntz 5h 52n+2 5 m2) cling ) Give) şi I 1+ 3/n 1+2n z 52 Ito, I. I to 1 +0 th th <I So By Ratiortent Elanl convergent. n=1 Ź an is absolutely convergent. n=1 (2) (-Engh In n=1 n clet an (6nga h7n

(-13" (62) nan n 6n lanl= | care cond") - n7 Ghin. It 6n Now, lanlyn n7 ht dim 6h = cim 6 cl clim lanlyn nes n7 nas n6 horo Cauchy Root tent Eland in convergent. no So, by z) Ź an éis absolutely convergent. nal (3) Ý Gu" n? In (n+2) n=1 (1) n-2 dn (112) chet an n² In (nt) tire mezej a a 19m) a nadim Now, clay (+2) > (n+233 in and dog sex n² log(n+2 je > n² LI n²log (htz) n2 By Comparison text, Since there in convergent. Therfere, is convergent h²log (ht2) a Ean is absolutely convergent El

(46) E ell ch an (no) clim lanlyn lim (hion he clim (nejh 6h² h clim (no) 6h elim forin och clayo (by L. Harpital) <I So by Cauchy Poot text Eland is convergent. > E an in absolutely convergut (4) E (nt.)" clim (on)in clim 6h cline for you htc noo no ( So Eland in divergent n (6) h7 8-1004 mil lim clim ht 8- lont h7 ant clim 8-lont

elim polynomial of deg 7 boolynomial of deg 4 (Applying L- Hospital Rule 4 times sl So by Root test E an is divergent.