7.5 Tests of convergence 205
The sum of the infinite series is then the limit of the sequence of partial sums. On the
other hand, the example of the harmonic series shows that when such a formula is not
known less direct procedures must be followed.
Given a series
a necessary first condition for convergence is that the limit of the sequence {a
r}be
zero:
a
r1 → 1 0asr 1 → 1 ∞
If this condition is satisfied, the series can be tested for convergence in a number of
ways.
The comparison test
Let
A 1 = 1 a
11 + 1 a
21 +1-1+ 1 a
r1 +1-
B 1 = 1 b
11 + 1 b
21 +1-1+ 1 b
r1 +1-
be two series of positiveterms. Then:
(i) If series B converges, then series Aconverges ifa
r1 ≤ 1 b
r.
(ii) If series B diverges, then series Adiverges ifa
r1 ≥ 1 b
r.
EXAMPLE 7.8The series
converges ifp 1 > 11 and diverges ifp 1 ≤ 11.
(i)p 1 = 11 : Sis the harmonic series, and diverges.
(ii)p 1 < 11 : each term of S(after the first) is larger than the corresponding terms of the
harmonic series, and Sdiverges.
(iii)p 1 > 11 : write the series as
= 111 + 1 s
11 + 1 s
21 +1-
S
pp pppp=+ ++
++++
1 +
1
2
1
3
1
4
1
5
1
6
1
7
S
ppp=+ + + + 1
1
2
1
3
1
4
rraaaa
=∑
=+++
1123∞