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Homelimit of $(1 + s_n)^n$ is one when limit of $n s_n$ is zero

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

The inequalities for differences of powers may be used to show that $\lim_{{n\to\infty}}(1+s_{n})^{n}=1$ when $\lim_{{n\to\infty}}ns_{n}=0$. 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.

$ns_{n}\leq(1+s_{n})^{n}-1\leq{ns_{n}\over 1-(n-1)s_{n}}$ |

Since $\lim_{{n\to\infty}}ns_{n}=0$, there must exist $N$ such that $ns_{n}<1/2$ when $n>N$. Hence, when $n>N$,

$|(1+s_{n})^{n}-1|<2|ns_{n}|$ |

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

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Proof

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## Mathematics Subject Classification

26D99*no label found*

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