: Generalized series

6.5 Generalized series

Generalized series with positive terms

Definition 240

[0FW]Let $I$ be an infinite family of indices and let $a_{i} : I \to [0, \infty]$ be a generalized sequence, we define the sum $\sum_{i \in I} a_{i}$ as

\sum_{i \in I} a_{i} = sup {\sum_{i \in K} a_{i} : K \in P_{f} (I)}

where $P_{f} (I)$ is the set of finite subsets $K \subseteq I$ .

E240

[0FX]Prerequisites:1.Note:From the written exam of March 27, 2010..Say for which $𝛼 \in ℝ$ the series

\sum_{(m, n) \in N^{2}} \frac{1}{(n + m + 1)^{𝛼}} .

converges. Then discuss, for $N \geq 3$ , the convergence of

\sum_{(m_{1}, \dots m_{N}) \in N^{N}} \frac{1}{(1 + m_{1} + \dots + m_{N})^{𝛼}} .

Hidden solution: [UNACCESSIBLE UUID ’0FY’]

E240

[0FZ]Let $I$ be a family of indices, let $a_{i}$ be a sequence with $a_{i} \geq 0$ ; let moreover $F$ be a partition of $I$ (not necessarily of finite cardinality); then prove that

\sum_{F \in F} \sum_{i \in F} a_{i} = \sum_{i \in I} a_{i} .

E240

[0G0]Difficulty:*. Let $I$ be a family of indices; let $a_{i, j} : I \times ℕ \to [0, \infty]$ a generalised succession, such that $j \mapsto a_{i, j}$ is weakly increasing for every fixed $i$ ; prove that

\sum_{i \in I} lim_{j \to \infty} a_{i, j} = lim_{j \to \infty} \sum_{i \in I} a_{i, j} .

(This is a version of the well-known Monotone convergence theorem).

Hidden solution: [UNACCESSIBLE UUID ’0G2’] [UNACCESSIBLE UUID ’0G1’]

[0G3]Extend the previous 3, replacing

ℕ

with a set of indexes

J

endowed with filtering ordering

\leq

[UNACCESSIBLE UUID ’0G4’]