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.

Details

The five series without rearrangement are
∞
∑
n=1
n+1
(-1)
n=ln2≈0.693147
,
∞
∑
n=1
n+1
(-1)
n=
1
2
ln2(ln2-γ)≈-0.159869
,
∞
∑
n=1
sinn/n=
1
2
(-1+π)≈1.0708
,
∞
∑
n=1
n+1
(-1)
sinn/n=
1
2
,
∞
∑
n=1
n+1
(-1)
n=
1
2
2
γ
1
2
-
2
γ
1
2
,
3
2
≈0.395101
,
where
γ≈0.577216
is Euler's constant,
ζ(s)
is the Riemann zeta function, and
ζ(s,a)
is the generalized Riemann zeta function.

External Links

Riemann Series Theorem (Wolfram MathWorld)
Conditional Convergence (Wolfram MathWorld)
Euler-Mascheroni Constant (Wolfram MathWorld)
Riemann Zeta Function (Wolfram MathWorld)

Permanent Citation

Victor Phan, Simon Tyler
​
​"Riemann's Theorem on Rearranging Conditionally Convergent Series"​
​http://demonstrations.wolfram.com/RiemannsTheoremOnRearrangingConditionallyConvergentSeries/​
​Wolfram Demonstrations Project​
​Published: October 15, 2013