Sequences and Series(a)
an { 0
lim - =
n->oo n 00
1. lnn
(b) 1m --= 0
n~oo n
ifO<a:::;l
if a> 1
*20. If an > 0 and limn-.oo(an+i/an) = r =/= 0 prove that
lim (an)Ifn = r
n->ooUse the result to show that limn_, 00 (n!)^1 /n /n = l/e.
185- If X1 = ,,/2 and Xn+I = J2+Fn (n = 1,2, ... ), prove that {xn}
converges, and find lim x n approximated to two decimal places. Hint:
Show that Xn < 2 for all n, and apply the proposition in problem 17.
4.6 Series of Complex Numbers
Definition 4.8. Let {an} be an infinite sequence of complex numbers. By
an infinite series of complex numbers (or briefly, a numerical series) whose
terms are those of the sequence {an} is understood a sum of the form
00
L an = a1 + a2 + · · · + an + · · ·
n=l
to which a meaning is attached in the following manner: Letting( 4.6-1)( 4.6-2)the sequence {Sn} is called the sequence of the partial sums of the se-
ries (4.6-1), and the series is said to be convergent or divergent according
to whether the sequence {Sn} converges or diverges. In the case of con-vergence, if limn-.oo Sn = S we call S the sum of the infinite series, or say
that the series converges to S, and write
oo n
S = "'an ~ = ai + a2 + · · · + an + · · · = n---+oo~ lim "'ak
n=l k=l
In the case of divergence, no sum or value is assigned to the series, except
when the sequence {Sn} is unbounded, in which case it may be said that
the series diverges to oo. However, if (<C*, x) is the underlying space, it
would be more proper to refer to this case as convergence to oo.If the sequence {Sn} is oscillatory, no sum is assigned to (4.6-1). Yet
by modifying the definition above of the sum of a series, it is possible to
assign a sum to certain classes of divergent series. There are several ways in
which this can be accomplished, leading to the modern theory of divergent
sen es.