# proof of Simultaneous converging or diverging of product and sum theorem

From the fact that $1+x\leq e^{x}$ for $x\geq 0$ we get

 $\sum_{n=1}^{m}a_{n}\leq\prod_{n=1}^{m}(1+a_{n})\leq e^{\sum_{n=1}^{m}a_{n}}$

Since $a_{n}\geq 0$ 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 ProofOfSimultaneousConvergingOrDivergingOfProductAndSumTheorem 2013-03-22 13:35:57 2013-03-22 13:35:57 Johan (1032) Johan (1032) 5 Johan (1032) Proof msc 30E20