proof of infinite product of sums 1\tmspace-.1667em+\tmspace-.1667emai result without exponentials

In this entry, we show how the proof presented in the parent entry may be modified so as to avoid use of the exponential functionDlmfDlmfMathworld. This modification makes it more elementary by not requiring that one first develop the theory of the exponential function before proving this basic result about infinite products. Note that it is only necessary to redo the part of the result which states that, if the series converges, then the product also converges because the proof of the opposite implication did not involve the exponential function.

We begin with a simple inequalityMathworldPlanetmath. Suppose that a and b are real numbers such that 0a and 0b<1/2. Then we have 2aba, hence

(1+a)(1+2b) =1+2b+a+2ab

Now suppose that the series a1+a2+a3+ converges to a value S. Since the convergence of an infinite series or product is not affected by removing a finite number of terms we may, without loss of generality, assume that S<1/2. Then, since the terms an are nonnegative for all n, for each partial sum sn we will have 0sn<1/2.

Clearly, t11+2s1. Suppose that, for some n, we have tn1+2sn. Then, using the definitions of tn and sn along with the inequality demonstrated above, we conclude that

tn+1 =tn(1+an+1)

Hence, if tn1+2sn, then tn+11+2sn+1 as well. By induction, we conclude that tn1+2sn for all n.

Thus, for all n, we have tn1+2sn1+2S. Substituting this inequality for the inequality tnesneS in the parent entry, the rest of the proof proceeds in exactly the same manner.

Title proof of infinite product of sums 1\tmspace-.1667em+\tmspace-.1667emai result without exponentialsMathworldPlanetmathPlanetmath
Canonical name ProofOfInfiniteProductOfSums1aiResultWithoutExponentials
Date of creation 2013-03-22 18:40:38
Last modified on 2013-03-22 18:40:38
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 11
Author rspuzio (6075)
Entry type Result
Classification msc 40A20
Classification msc 26E99