WOLFRAM|DEMONSTRATIONS PROJECT

Rearranging the Alternating Harmonic Series

​
target for series sum
0.693147
approximate number of sign changes
10
25
50
100
200
show rearranged series
show partial sums
show sign changes
Start of the rearranged series: 1-
1
2
+
1
3
-
1
4
+
1
5
-
1
6
+
1
7
-
1
8
+
1
9
-
1
10
+
1
11
-
1
12
+
1
13
-
1
14
+
1
15
-
1
16
+
1
17
-
1
18
+
1
19
-
1
20
+
1
21
-
1
22
+
1
23
-
1
24
+···
The partial sums just before the sign changes: {1,0.5,0.8333,0.5833,0.7833,0.6167,0.7595,0.6345,0.7456,0.6456,0.7365,0.6532,0.7301,0.6587,0.7254,0.6629,0.7217,0.6661,0.7188,0.6688,0.7164,0.6709,0.7144,0.6727}
Number of terms that are positive, negative, positive, etc.: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Terms in the alternating harmonic series (or any other conditionally convergent series) can be rearranged to achieve any desired real number as the sum. This Demonstration allows you to control the target sum and various options to see (part of) the rearranged series, the partial sums, and the pattern of sign changes. By default, the target is set to
log(2)≈0.693147…
.