Given any series
∑
a
n
, we define a series
∑
a
n
+
whose terms are all the positive terms of
∑
a
n
and a series
∑
a
n
−
whose terms are all the negative terms of
∑
a
n
. To be specific, we let
a
n
+
=
a
n
+
|
a
n
|
2
a
n
−
=
a
n
−
|
a
n
|
2
Notice that if
a
n
>
0
, then
a
n
+
=
a
n
and
a
n
−
=
0
, whereas if
a
n
<
0
, then
a
n
−
=
a
n
and
a
n
+
=
0
.
(a) If
∑
a
n
is absolutely convergent, show that both of the series
∑
a
n
+
and
∑
a
n
−
are convergent.
(b) If
∑
a
n
is conditionally convergent, show that both of the series
∑
a
n
+
and
∑
a
n
−
are divergent.