Question

(35 pls) Determine if each of the following seven series converge or diverge. Do not use a test for convergence or a test for divergence more than once. If you use the Integral Test, do not bother to show me that you checked the three prerequisites to that test. i) 2(04-) n.1 ii) (-1)"n(2n)! -1 (n + 1) (2n) 21 n=1 n 3 iii) 2+ n-1 iv) n=0 1+en 00 1 v) n-07T +n* vi) (-1)"e" 1+en n=0 vii) Ž(-1)" tan" (n) nad

Answer 1

ů Å (e Ym3 -1) her 993-1 an - e 31 + -1 1 + n3 3 Į + t. nad! +(hazitle (13) 927 (ngj33! let Ebn = {013 t. 13)221' (13)33! h + + n3 an lim n → lim bn n7% h3 t. t n> is E an lim (1+ (23) 21. (63)?3! affinite) Eh 3 is convergent convergent (prest (prest series) By limit comparison test & bn is convergent => convegent (-1)^n (2n)! (n+1) (20) n (2n)! (n+) (2n)an dim n (2n). n 2n. An-1).(2n-2) M8 (2 zn an antern 1 lim n7oo in f) (an) 2n nyo [n+1) (an) en

= lim (1-7)(1-2) (-3) - an) n700 I+I an - anti ant2 29 n(en)! (n+1) (2n+2)! (n+1) (2n)? (n +2) (2n+2) n (an)!(n+2) (2n+2) 25+ -(n+D (2n+2)! (n + 1)(2n) 2n (n+1)(n+2) (2n) 2n cent2)2n an anti > o Hence an = 0 an > anti & non is monotonically decreasing an decreasing montonically & dim an (Leibnizs test) Elonn (2n)! is na n+1) (an)? convergent . (2+ )" (2+)? wing integral pest (iii) an = têm (2+2)" an lim nyoo = n70 = Hence #o (2+2 / 2n is divergent hal

ż en Itean hto en Hin) = The an using integral test f(x) is continuous Iteas 0 ex da iteax let ex=t ex doc dt dt Itt [ tonit) tanle tanto 51 fly = 을 Il tinite) f(s) doc converges hence 을 en tean is converges neo