# limit of ${(1+{s}_{n})}^{n}$ is one when limit of $n{s}_{n}$ is zero

The inequalities for differences of powers may be used to show that
${lim}_{n\to \mathrm{\infty}}{(1+{s}_{n})}^{n}=1$ when ${lim}_{n\to \mathrm{\infty}}n{s}_{n}=0$.
This fact plays an important role in the development of the theory of the
exponential function^{} as a limit of powers.

To derive this limit, we bound $1+{s}_{n}$ using the inequalities for differences of powers.

$$n{s}_{n}\le {(1+{s}_{n})}^{n}-1\le \frac{n{s}_{n}}{1-(n-1){s}_{n}}$$ |

Since ${lim}_{n\to \mathrm{\infty}}n{s}_{n}=0$, there must exist $N$ such that $$ when $n>N$. Hence, when $n>N$,

$$ |

so, as $n\to \mathrm{\infty}$, we have ${(1+{s}_{n})}^{n}\to 1$.

Title | limit of ${(1+{s}_{n})}^{n}$ is one when limit of $n{s}_{n}$ is zero |
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Canonical name | LimitOf1SnnIsOneWhenLimitOfNSnIsZero |

Date of creation | 2013-03-22 15:48:55 |

Last modified on | 2013-03-22 15:48:55 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 7 |

Author | rspuzio (6075) |

Entry type | Proof |

Classification | msc 26D99 |