proof of Simultaneous converging or diverging of product and sum theorem

From the fact that $1+x\le e^x$ for $x\ge 0$ we get

$$\sum _{n=1}^m{a}_n\le \prod _{n=1}^m(1+a_n)\le e^{{\sum }_{n=1}^m{a}_n}$$

Since $a_n\ge 0$ both the partial sums and the partial products are monotone increasing with the number of terms. This concludes the proof.

Title	proof of Simultaneous converging or diverging of product and sum theorem
Canonical name	ProofOfSimultaneousConvergingOrDivergingOfProductAndSumTheorem
Date of creation	2013-03-22 13:35:57
Last modified on	2013-03-22 13:35:57
Owner	Johan (1032)
Last modified by	Johan (1032)
Numerical id	5
Author	Johan (1032)
Entry type	Proof
Classification	msc 30E20