infinite product of sums 1\tmspace-.1667em+\tmspace-.1667emai

Lemma.  Let the numbers ai be nonnegative reals.  The infinite product

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

converges iff the series  a1+a2+a3+  is convergentMathworldPlanetmathPlanetmath.

Proof.  Denote

i=1nan:=sn,i=1n(1+ai):=tn  (n=1, 2,).

Now  1+ai 1+ai1!+ai22!+=eai,  whence

tni=1neai=esn. (2)

We can estimate also downwards:

tn=(1+a1)(1+a2)(1+an)= 1+i=1nai++a1a2an>i=1nai=sn (3)

If the series is convergent with sum S, then by (2),


and since the monotonically nondecreasing sequenceMathworldPlanetmatht1,t2,t3,  thus is bounded from above, it converges (cf. limit of nondecreasing sequence).  So (1) converges.

If, on the other hand, the series is divergent, then  limnsn=  and by (3), also  limntn=,  i.e. the (1) diverges.

Title infinite product of sums 1\tmspace-.1667em+\tmspace-.1667emai
Canonical name InfiniteProductOfSums1ai
Date of creation 2013-03-22 18:40:01
Last modified on 2013-03-22 18:40:01
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 40A20
Classification msc 26E99
Related topic LimitOfRealNumberSequence
Related topic DeterminingSeriesConvergence
Related topic InfiniteProductOfDifferences1A_i
Related topic AbsoluteConvergenceOfInfiniteProductAndSeries