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单词 AbsolutelyConvergentInfiniteProductConverges
释义

absolutely convergent infinite product converges


Theorem.  An absolutely convergent (http://planetmath.org/AbsoluteConvergenceOfInfiniteProduct) infinite product

ν=1(1+cν)=(1+c1)(1+c2)(1+c3)(1)

of complex numbersPlanetmathPlanetmath is convergent.

Proof.  We thus assume the convergence of the product (http://planetmath.org/Product)

ν=1(1+|cν|)=(1+|c1|)(1+|c2|)(1+|c3|)(2)

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|<εp+

when  n certain nε.  Then we see that

|(1+cn+1)(1+cn+2)(1+cn+p)-1|=|1+ν=n+1n+pcν+μ,νcμcν++cn+1cn+2cn+p-1|
1+ν=n+1n+p|cν|+μ,ν|cμ||cν|++|cn+1||cn+2||cn+p|-1
=|(1+|cn+1|)(1+|cn+2|)(1+|cn+p|)-1|<ε  p+

as soon as  nnε.  I.e., the infinite product (1) converges, by the same convergence condition.

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更新时间:2025/5/4 15:07:30