4. 1 • SEQUENCES ANO SERIES 129
- EXAMPLE 4.3 Show that the series f i+ir;f;-^1 r f [;t.z +iC-~t] is
n=l n = l
convergent.
00 00 ,_,, ..
Solution Recall that the real series I:; ;!-z and I:; ~ a re convergent.
n=l n=l
Hence, Theorem 4.4 implies that the given complex series is convergent.
•EXAMPLE 4.4 Show that the series n~t (-t~n+i = n~i [<-~t +i~] is di-
vergent.
00
Solution We know that the real series I:; ~ is divergent. Hence, Theorem 4.4
n.= 1
implies that the given complex series is divergent.
00
- EXAMPLE 4.5 Show that the series I:; (1 +it is divergent.
n=l
Solution Here we set Zn = (1 + i}" and observe that Jim lznl = lim ( v'2)" =
n-+oo n-oo
oo. Thus lim Zn :/= 0, and Theorem 4.5 implies t hat the series is not convergent;
n~oo
hence it is divergent.
