Riemann's Theorem on Rearranging Conditionally Convergent Series

term of alternating series

-1

n+1

)\),

/ n

target value x

0.693147

number of terms

30

When

t

n

=

n+1

(-1)

n

,

∞

∑

n=1

t

n

 log(2) ≃ 0.693147.

x

=

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

+

1

25

-

1

26

+

1

27

-

1

28

+

1

29

+…

Conditionally convergent series of real numbers have the interesting property that the terms of the series can be rearranged to converge to any real value or diverge to

±∞

. In this Demonstration, you can select from five conditionally convergent series and you can adjust the target value

x

. The Demonstration rearranges the series, plots its

th

k

partial sum (the sum from 0 to the

th

k

term), and shows the rearranged series.