234 P-Summable
A property of a sequence under which the series of absolute values raised to a parameter power converges that is used to identify members belonging to a matching set of sequences.
definition [d] (\(p\)-Summable) From Kreyszig 1.2-3: a sequence \(x = (\xi_{j})\) of numbers is an element of \(\ell^{p}\) when
- \(\displaystyle\sum_{j=1}^{\infty} |\xi_{j}|^{p} < \infty\)
for a fixed real number \(p \geq 1\).
where
- \(x = (\xi_{j})\) is a sequence of scalars.
- \(p \geq 1\) is a fixed real number.
234.1 Elementary Example
234.1.1 Simple
A sequence is \(p\)-summable when \(\sum |\xi_{j}|^{p}\) is finite. For \(p = 1\), the finite sequence \((1,2,3)\) is \(1\)-summable.
\[ x = (1,2,3),\quad p = 1 \]
\[ \sum_{j=1}^{3} |\xi_{j}|^{1} = 6 < \infty \]
where
- \(p\)-summable means membership in \(\ell^{p}\) after zero-padding to an infinite sequence.
- \(p \geq 1\) is fixed.
234.1.2 General
For \(p = 2\), the infinite sequence \(\xi_{j} = 1/j\) is \(2\)-summable because \(\sum 1/j^{2}\) converges.
\[ x = \left(1,\ \dfrac{1}{2},\ \dfrac{1}{3},\ \ldots\right),\quad p = 2 \]
\[ \sum_{j=1}^{\infty} \left|\dfrac{1}{j}\right|^{2} < \infty \]
where
- \(x = (\xi_{j})\) is the sequence.
- finiteness of the sum is Kreyszig’s \(\ell^{p}\) membership test.