Question

1. True or false: (15 pts) {(-1)" tan (TC/2-3/n} is oscillating. (b) 1/2-1/4+1/6-1/8+1/10-..... converges conditionally. A convergent sequence is always Cauchy. {1/n) is a Cauchy sequence. (1-3)-(1-31/2)+(1-313)-(1-314 )+.....diverges. 2. Find limit sup and limit inf of the following sequences: (10 pts) (a){c+4) sin ng (b) {(1+m+)"} Limsup= limsup= Lmitinf= liminf= 3. Prove that either the following sequence has a limit or not. (20 pts) (a) 2n (b) n2+4n+2 n+6vn n-1

All of question 2 please

Answer 1

(a) { ( 2 + 4) sin nn} . 6 Ji 5 7 s .. 1 in sin no : CE, 58,0,4+4)(-5), (**)(-5),0, +4), 6+4)2,0, -... } dim supan = tim (om) time (le +4) ) = 453 = 253 lim inf xn = lim (in nikm) = lim ((+)(- 9)) nyo .-253 44-18) = -883 Limsup = 253 Lim inf = -253 3 Scanned with CamScanner

(0) { (i++)"} n + e lim (9) ht things (1+1)" - ht .se Slim (+ f(n)) (N) e lima fin) gin) { (nyos where lim ntas f(n)=0 > when limit exist .ntos limit. = lim gin) = 0 lim then Limsup and (1 + x) = é jmine is equal to Lim sup = Lim inf = lim se Lim sup & e Lim inf=e nyos cs Scanned with CamScanner