# inequalities for differences of powers

Oftentimes, one needs to estimate differences of powers of real numbers.
The following inequalities^{} are useful for this purpose:

$$ |

$$nx\le {(1+x)}^{n}-1$$ |

$${(1+x)}^{n}-1\le \frac{nx}{1-(n-1)x}$$ |

Here $n$ is an integer greater than 1. The first inequality holds when $$, the second inequality holds when $$, and the third inequality holds when $$. Equality can only occur in the latter two inequalities when $x=0$.

Title | inequalities for differences of powers |
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Canonical name | InequalitiesForDifferencesOfPowers |

Date of creation | 2013-03-22 15:48:42 |

Last modified on | 2013-03-22 15:48:42 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 11 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 26D99 |

Related topic | BernoullisInequality |