8.4. ALTERNATING SERIES
499
convergent series. Before we begin, we remind ourselves that
∞
k=1
(−1) k+1
k
= ln(2),
a fact which will be verified in a later section.
(a) First we make a quick analysis of the positive and negative terms of the
alternating harmonic series.
(i) Show that the series
∞
k=1
1
2k
diverges.
(ii) Show that the series
∞
k=1
1
2k + 1
diverges.
(iii) Based on the results of the previous parts of this exercise, what can we
say about the sums
∞
k=C
1
2k
and
∞
k=C
1
2k + 1
for any positive integer C? Be
specific in your explanation.
(b) Recall addition of real numbers is commutative; that is
a + b = b + a
for any real numbers a and b. This property is valid for any sum of finitely
many terms, but does this property extend when we add infinitely many terms
together?
The answer is no, and something even more odd happens. Riemann’s Theorem (after the nineteenth-century mathematician Georg Friedrich Bernhard
Riemann) states that a conditionally convergent series can be rearranged to
converge to any prescribed sum. More specifically, this means that if we choose
any real number S, we can rearrange the terms of the alternating harmonic
series
∞
k=1
(−1) k+1
k
so that the sum is S. To understand how Riemann’s Theorem works, let’s assume for the moment that the number S we want our
rearrangement to converge to is positive. Our job is to find a way to order the
sum of terms of the alternating harmonic series to converge to S.
(i) Explain how we know that, regardless of the value of S, we can find a
partial sum P 1
P 1 =
n 1
k=1
1
2k + 1
= 1 +
1
3
+
1
5
+ · · · +
1
2n 1 + 1
of the positive terms of the alternating harmonic series that equals or
499
convergent series. Before we begin, we remind ourselves that
∞
k=1
(−1) k+1
k
= ln(2),
a fact which will be verified in a later section.
(a) First we make a quick analysis of the positive and negative terms of the
alternating harmonic series.
(i) Show that the series
∞
k=1
1
2k
diverges.
(ii) Show that the series
∞
k=1
1
2k + 1
diverges.
(iii) Based on the results of the previous parts of this exercise, what can we
say about the sums
∞
k=C
1
2k
and
∞
k=C
1
2k + 1
for any positive integer C? Be
specific in your explanation.
(b) Recall addition of real numbers is commutative; that is
a + b = b + a
for any real numbers a and b. This property is valid for any sum of finitely
many terms, but does this property extend when we add infinitely many terms
together?
The answer is no, and something even more odd happens. Riemann’s Theorem (after the nineteenth-century mathematician Georg Friedrich Bernhard
Riemann) states that a conditionally convergent series can be rearranged to
converge to any prescribed sum. More specifically, this means that if we choose
any real number S, we can rearrange the terms of the alternating harmonic
series
∞
k=1
(−1) k+1
k
so that the sum is S. To understand how Riemann’s Theorem works, let’s assume for the moment that the number S we want our
rearrangement to converge to is positive. Our job is to find a way to order the
sum of terms of the alternating harmonic series to converge to S.
(i) Explain how we know that, regardless of the value of S, we can find a
partial sum P 1
P 1 =
n 1
k=1
1
2k + 1
= 1 +
1
3
+
1
5
+ · · · +
1
2n 1 + 1
of the positive terms of the alternating harmonic series that equals or