114 3 Real Analysis
339.Show that the sequence
√
7 ,
√
7 −
√
7 ,
√
7 −
√
7 +
√
7 ,
√
7 −
√
7 +
√
7 −
√
7 ,...
converges, and evaluate its limit.There is a vocabulary for translating the language of derivatives to the discrete frame-
work of sequences. The first derivative of a sequence(xn)n, usually called the first dif-
ference, is the sequence( xn)ndefined by xn=xn+ 1 −xn. The second derivative, or
second difference, is 2 xn= ( xn)=xn+ 2 − 2 xn+ 1 +xn. A sequence is increasing
if the first derivative is positive; it is convex if the second derivative is positive. The
Cesàro–Stolz theorem, which we discuss below, is the discrete version of L’Hôpital’s
theorem.
The Cesàro–Stolz Theorem.Let(xn)nand(yn)nbe two sequences of real numbers with
(yn)nstrictly positive, increasing, and unbounded. If
lim
n→∞xn+ 1 −xn
yn+ 1 −yn=L,
then the limit
nlim→∞xn
ynexists and is equal toL.
Proof.We apply the same-δargument as for L’Hôpital’s theorem. We do the proof
only forLfinite, the casesL=±∞being left to the reader.
Fix>0. There existsn 0 such that forn≥n 0 ,
L−2
<
xn+ 1 −xn
yn+ 1 −yn<L+
2
.
Becauseyn+ 1 −yn≥0, this is equivalent to
(
L−
2
)
(yn+ 1 −yn)<xn+ 1 −xn<(
L+
2
)
(yn+ 1 −yn).We sum all these inequalities fornranging betweenn 0 andm−1, for somem. After
cancelling terms in the telescopic sums that arise, we obtain
(
L−
2
)
(ym−yn 0 )<xm−xn 0 <(
L+
2
)
(ym−yn 0 ).We divide byymand write the answer as