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单词 InfiniteProductOfDifferences1ai
释义

infinite product of differences 1\\tmspace-.1667em-\\tmspace-.1667emai


We consider the infinite products of the form

i=1(1-ai)=(1-a1)(1-a2)(1-a3)(1)

and the series  a1+a2+a3+  where the numbers ai are nonnegative reals.

  • If the series converges, then also the product converges and has a value which does not depend on the order of the factors and vanishes only when some of the factors is 0.

  • If  limiai=0  but the series diverges, then the value of the infinite product is always zero though no of the factors were zero.

Example.(1-12)(1-13)(1-14)= 0;  see the harmonic seriesMathworldPlanetmath.

Proof.1.  Now we have  limiai=0  (see the necessary condition of convergence of series), and so  ai<12  when  ii0.  We write

i=1(1-ai)=i=1i0-1(1-ai)i=i0(1-ai)(2)

and set in the last product

1-ai=111-ai=11+ai1-ai,

whence

i=i0n(1-ai)=1i=i0n(1+ai1-ai).(3)

As  ai<12, we have  11-ai<2  and thus 0<ai1-ai<2ai,  and therefore the seriesi=i0ai1-ai with nonnegative is absolutely convergent.  The theorem of the http://planetmath.org/node/6204parent entry then says that the product in the denominator of the right hand side of (3) tends, as  n,  to a finite non-zero limit, which don’t depend on the order of the factors.  Consequently, the same concerns the product of the left hand side of (3).  By (2), we now infer that the given product (1) converges, its value is on the order and it vanishes only along with some of its factors.
2.  There is an i0 such that  ai<1  when  ii0,  whence  ai1-ai>ai  and the series  i=i0ai1-ai diverges.  The denominator of the right hand side of (3) tends, as  n,  to the infinity and thus the product of the left hand side to 0.  Hence the value of (1) is necessarily 0, also when all factors were distinct from 0.

References

  • 1 E. Lindelöf: Differentiali- ja integralilaskuja sen sovellutukset III.2.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).
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更新时间:2025/5/4 5:18:42