474
8.3. SERIES OF REAL NUMBERS
⊳
The result of Activity 8.8 is the following important conditional statement:
If the series
∞
k=1
a k converges, then the sequence {a k } of kth terms converges
to 0.
It is logically equivalent to say that if the sequence {a k } of n terms does not converge to
0, then the series
∞
k=1 a k cannot converge. This statement is called the Divergence Test.
The Divergence Test. If lim k→∞ a k 0, then the series
a k diverges.
Activity 8.9.
Determine if the Divergence Test applies to the following series. If the test does not
apply, explain why. If the test does apply, what does it tell us about the series?
(a)
k
k+1
(b)
(−1) k
(c)
1
k
⊳
Note well: be very careful with the Divergence Test. This test only tells us what
happens to a series if the terms of the corresponding sequence do not converge to 0. If
the sequence of the terms of the series does converge to 0, the Divergence Test does not
apply: indeed, as we will soon see, a series whose terms go to zero may either converge or
diverge.
The Integral Test
The Divergence Test settles the questions of divergence or convergence of series
a k
in which lim k→∞ a k 0. Determining the convergence or divergence of series
a k in
which lim k→∞ a k = 0 turns out to be more complicated. Often, we have to investigate the
sequence of partial sums or apply some other technique.
As an example, consider the harmonic series 5
∞
k=1
1
k
.
5 This series is called harmonic because each term in the series after the first is the harmonic mean of
the term before it and the term after it. The harmonic mean of two numbers a and b is
2ab
a+b . See “What’s
Harmonic about the Harmonic Series", by David E. Kullman (in the College Mathematics Journal, Vol. 32, No.
3 (May, 2001), 201-203) for an interesting discussion of the harmonic mean.
8.3. SERIES OF REAL NUMBERS
⊳
The result of Activity 8.8 is the following important conditional statement:
If the series
∞
k=1
a k converges, then the sequence {a k } of kth terms converges
to 0.
It is logically equivalent to say that if the sequence {a k } of n terms does not converge to
0, then the series
∞
k=1 a k cannot converge. This statement is called the Divergence Test.
The Divergence Test. If lim k→∞ a k 0, then the series
a k diverges.
Activity 8.9.
Determine if the Divergence Test applies to the following series. If the test does not
apply, explain why. If the test does apply, what does it tell us about the series?
(a)
k
k+1
(b)
(−1) k
(c)
1
k
⊳
Note well: be very careful with the Divergence Test. This test only tells us what
happens to a series if the terms of the corresponding sequence do not converge to 0. If
the sequence of the terms of the series does converge to 0, the Divergence Test does not
apply: indeed, as we will soon see, a series whose terms go to zero may either converge or
diverge.
The Integral Test
The Divergence Test settles the questions of divergence or convergence of series
a k
in which lim k→∞ a k 0. Determining the convergence or divergence of series
a k in
which lim k→∞ a k = 0 turns out to be more complicated. Often, we have to investigate the
sequence of partial sums or apply some other technique.
As an example, consider the harmonic series 5
∞
k=1
1
k
.
5 This series is called harmonic because each term in the series after the first is the harmonic mean of
the term before it and the term after it. The harmonic mean of two numbers a and b is
2ab
a+b . See “What’s
Harmonic about the Harmonic Series", by David E. Kullman (in the College Mathematics Journal, Vol. 32, No.
3 (May, 2001), 201-203) for an interesting discussion of the harmonic mean.