WOLFRAM|DEMONSTRATIONS PROJECT

Sum of the Alternating Harmonic Series (I)

​
n
5
The graph shows that:
The purple area is
11-
1
2
=1-
1
2
.
The blue area is
1
2
2
3
-
2
4
=
1
3
-
1
4
.
The green area is
1
4
4
5
-
4
6
+
1
4
4
7
-
4
8
=
1
5
-
1
6
+
1
7
-
1
8
.
The yellow area is
1
8
8
9
-
8
10
+
1
8
8
11
-
8
12
+
1
8
8
13
-
8
14
+
1
8
8
15
-
8
16
=
1
9
-
1
10
+
1
11
-
1
12
+
1
13
-
1
14
+
1
15
-
1
16
.
Therefore
1-
1
2
+
1
3
-
1
4
+
1
5
-
1
6
+…=
∞
∑
n=1
n+1
(-1)
1
n
=ln2
.