1549901369-Elements_of_Real_Analysis__Denlinger_

2.5 Monotone Sequences 101

limit. To analyze a sequence { Xn} for convergence, students are encouraged to

calculate a number of terms of the sequence and observe the trend. For exam-

ple, we could use a calculator to calculate terms of the sequence { ( 1 + ~) n},

as shown in Table 2.4.

Table 2.4

n 1 5 10 20 100 1,000 5,000

(l+~J 2 2 .488 2.594 2.653 2.705 2.717 2.718

As n gets larger and larger, the values of ( 1 + ~) n seem to be getting

closer to each other; we could conclude from the computed values in t he table

that { ( 1 + ~) n} converges and t hat its limit is a number close to 2. 718.

WARNING: Use of calculation to determine convergence and limits is
frequently unreliable. Uncritical reliance upon calculation can lead to quite
erroneous conclusions. The following example will demonstrate the dangers
inherent in this approach.
00
Recall from your calculus course that an infinite series L Xn is said to
n=l
converge to a sum S if and only if the sequence {Sn} of "partial sums" Sn =
n
L Xk converges to S. That is ,
k=l

oo n

'°"" ~ Xn = S ¢:> n--+oo lim '°"" ~ Xk = S.

n=l · k=l

Thus, convergence of a series L Xn is determined by the convergence of an
associated sequence {Sn}. The example we are going to give will be a series that
diverges, although its divergence is extremely slow-so slow that any calculator
or computer will be fooled into concluding that it converges and even produce
a finite number as its sum.

Example 2.5.16 The harmonic series

n 1

n--+oo lim '°"" L__; -k = +oo.
k=l

00 1

2:-n

n=l

diverges to + oo; that is,