Ratio Test

RT : Ratio Test Ean 0 V n Cim anti = s 8 an RT n- nl If pll Ean conv. A If p>I Ean div RT fails If P=1 Exp check Con / Div. = 8 2 I n Apply RT an = n? Do & E en n=1,7, en 2 n=1 (n+1) n+1 2 Pim and = lim e = lim (n+1) 2 n+1 now n2 e n2 an n-900 n-s 8 en n C = lim intan+l = n- 00 enc e

w n+c an V Apply RT O 6 £ 3 Inn Inn n n=2 90 3 a n+3 Inn lim 3 = 3 lim Inn lim anti a . In(n+1) In(n+1) n+2 n 00 an now n & 1 n =3 lim n+1 = 3 lim = 3(1) n n-300 I n a n+l 8 = 3 > 1 n + 2 Hence, { 3 div by RT n=2 1 "Div" since it is the harmonic Series 3 n Yan)o O & n=1,2, Apply RT n=1 1/5 lim anti = lin n+1 = lim n = 1 31 an n+1 n-soo n-s 00 n-s w RT fails to make decision so we try to apply another Test. n! D n! apply RT In = ) 0 & n=1,7,3 4 en n e n=l (mm) a Intu! n+1 = 8 >II A I 1.

your lim anti = lim (nty! n+1 = 8 >II = lim e 00 now an n-w ntt nt n- e Hence, E n! en div by RT n! = n (n+1)! = n! (n+1) =(n+1) nl 20 8 3 E n! Apply RT an so & 5 10" an n=1 n+1 6 TO lim and = lim = 1/6 lim (n+1)=00 o/1 10 n-so an n D 10th now 10 E n'i 10" div by RT ((n) = 10 n+1 n x f (n+1) 10 10 + 00 Apply Root Test V = Sa Vb 7 E(p) an >0 & n=1,2, -

7 2/2/1 3 , an >o v . n=1 3 A 3 Pim Van = lim x = lim n3 nso now "V n s ow \ 3 = 3 lim I = 3 lin 3 no) n-soo n lim () n-00 n-soo 3 = = 1/7 = 3 15)1 lim (vv) n 8 T English n3 div by Root Test 1 0 10.1" 8 In2 Root Test 15 E(1 1/1 n=1 the lim Van a3 = (1-1)30 n-300 n lim Va an>o for largn 10.1 Th, now 15 n2 lim (1- 1 S lim ( 1 - 1/2 = n-1 8 n-> = e i' = 1 511-19 8 Con by Root Test

8 Hence Er. - +S Con by Root Test n=1 n 30 00 Apply Root Test n=1 9-=4-11== an a2= lim v an anyo n-300 lim (th - -12- 0- -0=031 O = n-s 00 conu by Root Test Exp Recusive terms a, = 2 / 9 n+1 2= an - 8 anti = M/s an Does Ean an conv ? n=1 Pim antl = lim 2/4 = . (1) o Apply Ration n-soo an now Yes it does conv. by RT