The Real And Complex Number Systems

0 ≤ d 1 *
by Theorem 8.23 (i). Note that d n − fn is a positive increasing sequence, so we have
d 0. **
Since
T n 2 n − 2
which implies that
lim n→
n
2 n − ∑ 1/ k lim n→
a n 2 − d p.
k1
By (*) and (**), we have proved that 1 p 1.
Remark: (1) The use of Integral Test is very useful since we can know the behavior
of a given series by integral. However, in many cases, the integrand may be so complicated
that it is not easy to calculate. For example: Prove that the convergence of
∑
n2
1
nlog n p ,wherep 1.
Of course, it can be checked by Integral Test. But there is the Theorem called Cauchy
Condensation Theorem much powerful than Integral Test in this sense. In addition, the
reader can think it twice that in fact, Cauchy condensation Theorem is equivalent to
Integral Test.
(Cauchy Condensation Theorem)Let a n be a positive decreasing sequence. Then
∑
n1
a n converges if, and only if, ∑
k0
2 k a 2
k converges.
Note: (1) The proof is not hard; the reader can see the book, Principles of
Mathematical Analysis by Walter Rudin, pp 61-63.
(2) There is an extension of Cauchy Condensation Theorem (Oskar Schlomilch):
Suppose that a k be a positive and decreasing sequence and m k ⊆ N is a sequence. If
there exists a c 0 such that
0 m k2 − m k1 ≤ cm k1 − m k for all k,
then
∑
k1
a k converges if, and only if, ∑m k1 − m k a mk .
k0
Note: The proof is similar with Cauchy Condensation Theorem, soweomitit.
(2) There is a similar Theorem, we write it as a reference. If t ≥ a, ft is a
non-negative increasing function, then as x ≥ a, wehave
x
∑ fn − ftdt
a≤n≤x
a
≤ fx.
Proof: The proof is easy by drawing a graph. So, we omit it.
P.S.: The theorem is useful when we deal with some sums. For example,
ft log t.
Then