Riemann’s theorem on rearrangements

Riemann’s theorem on rearrangements

If the map

n

↦

n

′

is a bijection on

ℕ

, we say that the

sequence

(

a

n

′

)

is a rearrangement of

(

a

n

)

.

The following

theorem

, which is due to Riemann, shows that the

convergence

of a

conditionally convergent

series depends so much on the order of its terms; in particular, a conditionally convergent series can be made to

converge

to any

real number

by changing the order of its terms.

Theorem (

Riemann series theorem

).

Let

(

a

n

)

be a sequence in

ℝ

such that

∑

n

=

1

∞

a

n

converges but

∑

n

=

1

∞

|

a

n

|

=

∞

, i.e,

∑

a

n

is conditionally convergent. Let

-

∞

≤

α

<

β

≤

∞

be arbitrary. Then there exists a rearrangement

(

a

n

′

)

such that

lim inf

N

⁡

∑

n

′

=

1

N

a

n

′

=

α

and

lim sup

N

⁡

∑

n

′

=

1

N

a

n

′

=

β

.

Proof.

Let

a

n

+

=

max

⁡

{

0

,

a

n

}

and

a

n

-

=

min

⁡

{

0

,

-

a

n

}

. Then we have

a

n

=

a

n

+

-

a

n

-

and

|

a

n

|

=

a

n

+

+

a

n

-

. Since

∑

a

n

<

∞

, both

∑

a

n

+

and

∑

a

n

-

diverge or converge simultaneously. But since

∑

|

a

n

|

=

∞

, we see that at least one of

∑

a

n

+

and

∑

a

n

-

must diverge. It follows that

∑

a

n

+

=

∞

and

∑

a

n

-

=

∞

.

Also by the

n

th term test,

lim

n

⁡

a

n

+

=

lim

n

⁡

a

n

-

=

0

.

Now we pass to

subsequence

of

(

a

n

+

)

by removing all terms with

a

n

+

=

0

and

a

n

≠

0

. For

a

n

-

, we remove all terms with

a

n

-

=

0

. Let us denote the subsequences still as

(

a

n

+

)

and

(

a

n

-

)

. Since only zeros have been removed,

∑

a

n

+

and

∑

a

n

-

are still divergent.

Now we will define integers

m

j

and

k

j

for

j

∈

ℕ

, and consider the series

a

1

+

+

a

2

+

+

⋯

+

a

m

1

+

-

a

1

-

-

a

2

-

-

⋯

-

a

k

2

-

+

a

m

1

+

1

+

+

a

2

+

+

⋯

+

a

m

2

+

-

a

k

1

+

1

-

-

a

2

-

-

⋯

-

a

k

2

-

+

⋮

This series is clearly a rearrangement of

∑

a

n

, by our choice of the subsequences

a

n

+

and

a

n

-

.

We pick up two sequences

α

j

and

β

j

such that

α

j

→

α

,

β

j

→

β

,

α

n

≤

β

n

and

β

1

>

0

. We choose

m

1

such that

∑

n

=

1

m

1

≥

β

1

but

∑

n

=

1

m

1

-

1

<

β

1

. We choose

k

1

such that

∑

n

=

1

m

1

-

∑

n

=

1

k

1

≤

α

1

but

∑

n

=

1

m

1

-

∑

n

=

1

k

1

-

1

>

α

1

. We continue this way, inductively.

Since

lim

n

⁡

a

n

+

=

lim

n

⁡

a

n

-

=

0

, the subsequences of the sequence of partial sums that end with

a

m

j

+

and

a

k

j

-

converge to

β

and

α

, and it can be seen that no subsequence can be found with a limit larger than

β

or lower than

α

.

□

References

1

Rudin, W.,

Principles of

Mathematical Analysis

, McGraw Hill, 1976.

Title

Riemann’s theorem on rearrangements

Canonical

name

RiemannsTheoremOnRearrangements

Date of creation

2013-03-22 17:31:52

Last modified on

2013-03-22 17:31:52

Owner

Gorkem (3644)

Last modified by

Gorkem (3644)

Numerical id

23

Author

Gorkem (3644)

Entry type

Theorem

Classification

msc 40A05

Synonym

Riemann series theorem

Related topic

UnconditionallyConvergent

Related topic

FiniteChangesInConvergentSeries

Related topic

FiniteChangesInConvergentSeries2

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