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absolutely convergent infinite product
converges∗
pahio†
2013-03-22 2:20:16
Theorem. An absolutely convergent infinite product
∞
Y
(1+cν ) = (1+c1 )(1+c2 )(1+c3 ) · · ·
(1)
ν=1
of complex numbers is convergent.
Proof. We thus assume the convergence of the product
∞
Y
(1+|cν |) = (1+|c1 |)(1+|c2 |)(1+|c3 |) · · ·
(2)
ν=1
Let ε be an arbitrary positive number. By the general convergence condition
of infinite product, we have
|(1+|cn+1 |)(1+|cn+2 |) · · · (1+|cn+p |) − 1| < ε ∀ p ∈ Z+
when n = certain nε . Then we see that
|(1+cn+1 )(1+cn+2 ) · · · (1+cn+p ) − 1| = |1 +
n+p
X
cν +
X
ν=n+1
n+p
X
51+
ν=n+1
|cν | +
cµ cν + . . . + cn+1 cn+2 · · · cn+p − 1|
µ, ν
X
|cµ ||cν | + . . . + |cn+1 ||cn+2 | · · · |cn+p | − 1
µ, ν
= |(1+|cn+1 |)(1+|cn+2 |) · · · (1+|cn+p |) − 1| < ε
as soon as n = nε . I.e., the infinite product (1) converges, by the same convergence condition.
∗ hAbsolutelyConvergentInfiniteProductConvergesi
created: h2013-03-2i by: hpahioi version: h41439i Privacy setting: h1i hTheoremi h40A05i h30E20i
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1
∀ p ∈ Z+