Real Sequence (1 + x over n)^n is Convergent

Theorem
The sequence $\sequence {s_n}$ defined as:


 * $s_n = \paren {1 + \dfrac x n}^n$

is convergent.

Proof
From Cauchy's Mean Theorem:


 * $(1): \quad \displaystyle \paren {\prod_{k \mathop = 1}^n a_k}^{1/n} \le \frac 1 n \paren {\sum_{k \mathop = 1}^n a_k}$

for $r_1, r_2, \ldots, r_n$.

Setting:
 * $a_1 = a_2 = \ldots = a_{n - 1} := 1 + \dfrac x {n - 1}$

and:
 * $a_n = 1$

Substituting for $a_1, a_2, \ldots, a_n$ into $(1)$ gives:

The above is valid only if $a_1, a_2, \ldots, a_n$ are positive.

Hence we have shown that $\sequence {s_n}$ is increasing when:
 * $1 + \dfrac x {n - 1} \ge 0$

that is, when:
 * $n \ge 1 - x$

From Equivalence of Definitions of Real Exponential Function: Limit of Sequence implies Sum of Series, we have:


 * $(2): \quad \paren {1 + \dfrac x n}^n \le 1 + \size x + \dfrac {\size x^2} {2!} + \dotsb + \dfrac {\size x^n} {n!}$

Since there exists $N$ such that:
 * $\forall n > N: \dfrac {\size x^n} {n!} \le \paren {\dfrac 1 2}^n$

it follows from $(2)$ that:

Hence we have that $\sequence {s_n}$ is strictly increasing and bounded above.

So by the Monotone Convergence Theorem (Real Analysis), $\sequence {s_n}$ is convergent.

As $1 + \dfrac x n$ is positive when $n$ is large enough, it follows that the limit of $\sequence {s_n}$ is positive.