Chapter 3 - CBU

(3)1X( 1) n = 1 1 + 1 1 + · · · .n=0S = (s n ) = (1, 0, 1, 0, . . . ) diverges =)3.7. INTRODUCTION TO INFINITE SERIES 1471X( 1) n diverges.Theorem (3.7.3 —nth Term Test). If P x n converges, lim(x n ) = 0.Proof. P x n converges =) s = lim(s n ) exists =)s = lim(s n 1 ) =)lim(x n ) = lim(s n s n 1 ) = lim(s n ) lim(s n 1 ) = s s = 0.⇤Example.(4) Geometric series with |r| 1 diverges since (r n ) diverges.n=0(5) ForBut1Xn=11p = 1 + p 1 + p 1 + · · · + p 1 + · · · ,n 2 3 nlim(x n ) = lims n = 1 + 1 p2+ 1 p3+ · · · + 1 p n⇣ 1p n⌘= 0.Thus lim(s n )1p + p 1 + p 1 + · · · + p 1 = n ·n n n n| {z }n termslim( p 1X 1n) ! 1, so p diverges. nx=11p n= p nNote. This implies the converse of Theorem 3.7.3 is not true.