102 Chapter 2 • Sequences
n 1
Proof. \:/n E N, let Sn = "°"' ~k -. We want to prove that n-oo lim Sn = +oo.
k=lNow, \:/n EN, Sn+l =Sn + -
1
Sn· Thus, {Sn} is monotone increasing.
n+l
Hence, by Theorem 2.5.14 (a), we need only show that {Sn} is unbounded
above. Now, \:/n E N,
S2n = 1 + ~ + ( ~ + ~) + ( ~ + ~ + ~ + ~) + ... + ( 2n-; + 1 + ... + 2:)
1 + ~ + (~ + ~) + (~ + ~ + ~ + ~) + ... + (~ +... + ~)
2 4 4 8 8 8 8 2n 2n
1 1 1
= 1 + -+ - + · · · + - ( n + 1 terms; n ~ 's)
2 2 2
n
= 1+ 2·
Thus, the sequence {Sn} is unbounded above. Therefore, by Theorem 2.5.14
(a), lim Sn = +oo. D
n-oo
Lessons Drawn from Example 2.5.16: This example provides an excel-
lent demonstration of the folly of relying too heavily on calculators or computers
in finding limits. Try this one on your favorite calculator or computer. We have
just proved that the sequence {Sn} diverges to +oo. It would be expected,
therefore, that after not too many terms, Sn will exceed a small number, say
10. Try computing terms Sn for yourself. How many terms does it take to reach
10? If you actu ally try this by directly calculating Sn on your calculator, you
will probably give up in frustration. But please, read on.
By the "integral test" for infinite series used in elementary calculus, we
know that
n 1
ln(n + 1) < L k < 1 + ln n.
k=l
Thus, using a calculator to calculate ln(n + 1) and ln n we can see that
after 1,000 terms Sn h as not yet reached 8. It takes about 10 ,000 terms for Sn
to reach 10. After 100,000 terms, Sn is only about 12. After a million terms,
Sn is only about 14 , after a billion terms, Sn is about 20 , and after a trillion
terms, Sn is not yet 30! The sequence {Sn} is what we call a "slowly diverging"
sequence.
. 1
But even worse: eventually, say when n ?: n 0 , - is so small t hat the cal-
n
culator cannot distinguish it from 0. That is , \:/n ?: n 0 , the calculator uses 0
. 1
instead of -. Then \:/n ?: no, the calculator thinks Sn = Sno. Said another
n
00 1
way, the calculator thinks that the series L - converges, because it thinks
n=l n
the sequence {Sn} is eventually constant.