# 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 $(1+s_{n})^{n}\to 1$.

Title limit of $(1+s_{n})^{n}$ is one when limit of $ns_{n}$ is zero LimitOf1SnnIsOneWhenLimitOfNSnIsZero 2013-03-22 15:48:55 2013-03-22 15:48:55 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Proof msc 26D99