7.5 Generalized series
Generalized series with positive terms
[0FW]Let \(I\) be an infinite family of indices and let \(a_ i:I→[0,∞]\) be a generalized sequence, we define the sum \(∑_{i∈ I}a_ i\) as
where \({\mathcal P}_{\mathfrak f}(I)\) is the set of finite subsets \(K⊆ I\).
- E241
[0FX]Prerequisites:1.Note:From the written exam of March 27, 2010..Say for which \(𝛼∈ℝ\) the series
\[ ∑_{(m,n)∈{\mathbb N}^ 2}\frac{1}{(n+m+1)^𝛼}\quad . \]converges. Then discuss, for \(N≥ 3\), the convergence of
\[ ∑_{(m_ 1,\ldots m_ N)∈{\mathbb N}^ N}\frac{1}{(1+m_ 1+\ldots + m_ N)^𝛼}\quad . \]Hidden solution: [UNACCESSIBLE UUID ’0FY’]
- E241
[0FZ]Let \(I\) be a family of indices, let \(a_ i\) be a sequence with \(a_ i≥ 0\); let moreover \(\mathcal F\) be a partition of \(I\) (not necessarily of finite cardinality); then prove that
\[ ∑_{F∈\mathcal F}∑_{i∈ F} a_ i = ∑_{i∈ I} a_ i\quad . \]- E241
[0G0]Difficulty:*. Let \(I\) be a family of indices; let \(a_{i,j}:I× ℕ→ [0,∞]\) a generalised succession, such that \(j↦ a_{i,j}\) is weakly increasing for every fixed \(i\); prove that
\[ ∑_{i∈ I} \lim _{j→∞} a_{i,j} = \lim _{j→∞} ∑_{i∈ I} a_{i,j}~ ~ . \](This is a version of the well-known Monotone convergence theorem).
Hidden solution: [UNACCESSIBLE UUID ’0G2’] [UNACCESSIBLE UUID ’0G1’]