sum.htm Sums and Unconditional Summation

Sums and Unconditional Convergence

For manipulations with power series we are interested in infinite sums such as

∞
∑
j,k = 0

a_j,k.

We need to justify changing the order of summation and conditions which assure that

∞
∑
j,k = 0

a_j,k =

∞
∑
j=0

⎛
⎝

∞
∑
k=0

a_j,k

⎞
⎠

∞
∑
j=0

⎛
⎝

∞
∑
k=0

a_j,k

⎞
⎠

The condition which justifies the manipulations is absolute or unconditional convergence of the series.

Theorem. Suppose that all the finite [partial] sums

∑
j,k ∈ finite set
|a_j,k|

form a bounded set. Then there is a unique finite (complex) number

S = ∞
∑
j,k = 0
a_j,k

which may be calculated as

∞
∑
j,k = 0
a_j,k = ∞
∑
j=0
⎛
⎝ ∞
∑
k=0
a_j,k ⎞
⎠ = ∞
∑
j=0
⎛
⎝ ∞
∑
k=0
a_j,k ⎞
⎠ ,

or with any rearrangement in the summation of the terms - for example

S = ∞
∑
k=0
⎛
⎝ k
∑
j=0
a_k,k−j ⎞
⎠ .

Since the sum may be calculated in "any order", the convergence is called unconditional .

Notes on Unconditional Convergence

Let A be an index set¹. For each α ∈ A, Let z_α be a complex number. Then the sum

∑
α ∈ A

z_α

converges unconditionally to S if almost all the finite sums are close to S in the precise sense:

Given ϵ > 0, there is a finite set F_ϵ such that if F is any finite subset of A, F ⊇ F_ϵ, then

⎢
⎢

∑
α ∈ F

z_α − S

⎢
⎢

< ϵ.

Note that if ∑_{α ∈ A} z_α converges unconditionally to S, then ∑_{α ∈ A} ℜz_α converges unconditionally to ℜS and ∑_{α ∈ A} ℑz_α converges unconditionally to ℑS.

In particular, if z_α = x_α ≥ 0, then

S =

sup
F finite

∑
α ∈ F

x_α.

In this case ∑_{α ∈ A} x_α converges unconditionally iff the set of finite sums ∑_{α ∈ F} x_α, F finite, is bounded.

Let x_α be real. Define

x_α⁺

max

(x_α,0),

x_α⁻

max

(−x_α,0),

so that

x_α

= x_α⁺ − x_α⁻,

|x_α|

= x_α⁺ + x_α⁻.

Theorem. Let x_αbe real. The sum ∑_{α ∈ A}x_α converges unconditionally iff the sums ∑_{α ∈ A}x_α⁺, ∑_{α ∈ A}x_α⁻, and ∑_{α ∈ A}|x_α| all converge unconditionally.
Proof. There is a finite set F₁ such that |∑_{α ∈ F,F ⊇ F₁} x_α − S| < 1. Then if F is any finite set,

∑
α ∈ F

x_α⁺

≤

∑
α ∈ F\F₁

x_α⁺ +

∑
α ∈ F∩F₁

x_α⁺

≤ 1 +

∑
α ∈ F₁

|x_α|

≤ C.

Thus all the finite sums of ∑_{α ∈ A} x_α⁺, ∑_{α ∈ A} x_α⁻, and ∑_{α ∈ A} |x_α| are bounded.

Summary

By considering z_α = x_α⁺ − x_α⁻ + i (y_α⁺ − y_α⁻), we obtain:

1.: The sum ∑_{α ∈ A} z_α converges unconditionally iff
•: The sum ∑_{α ∈ A} |z_α| converges unconditionally.

iff

•: The set of finite sums ∑_{α ∈ F} |z_α|, F finite, is bounded.

2.

If {A_j} is a sequence of sets increasing to A ( A₀ ⊆ A₁ ⊆ A₂ ⊆ …, A = ∪_j=0^∞ A_j), it is easy to see that if ∑_{α ∈ A} z_α converges unconditionally, then

∑
α ∈ A

z_α =

lim
j → ∞

∑
α ∈ A_j

z_α.

3.

Rearrangement: If ∑_{α ∈ A} z_α converges unconditionally, and A is a disjoint union of sets, A = ∪_j=0^∞ B_j, with the B_j pairwise disjoint, then

∑
α ∈ A

z_α =

∞
∑
j=0

⎛
⎝

∑
α ∈ B_j

z_α

⎞
⎠

Footnotes:

¹For technical reasons, if A = ∅, the empty set, interpret ∑_{α ∈ ∅}z_α = 0.

File translated from T_EX by T_TH, version 4.03.
On 16 Nov 2013, 20:52.