# 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 |
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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 |