One Plus Reciprocal to the Nth
Theorem
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = \paren {1 + \dfrac 1 n}^n$.
Then $\sequence {x_n}$ converges to a limit as $n$ increases without bound.
Proof
First we show that $\sequence {x_n}$ is increasing.
Let $a_1 = a_2 = \cdots = a_{n - 1} = 1 + \dfrac 1 {n - 1}$.
Let $a_n = 1$.
Let:
- $A_n$ be the arithmetic mean of $a_1 \ldots a_n$
- $G_n$ be the geometric mean of $a_1 \ldots a_n$
Thus:
- $A_n = \dfrac {\paren {n - 1} \paren {1 + \dfrac 1 {n - 1} } + 1} n = \dfrac {n + 1} n = 1 + \dfrac 1 n$
- $G_n = \paren {1 + \dfrac 1 {n - 1} }^{\dfrac {n - 1} n}$
- $G_n \le A_n$
Thus:
- $\paren {1 + \dfrac 1 {n - 1} }^{\frac {n - 1} n} \le 1 + \dfrac 1 n$
and so:
- $x_{n - 1} = \paren {1 + \dfrac 1 {n - 1} }^{n - 1} \le \paren {1 + \dfrac 1 n}^n = x_n$
Hence $\sequence {x_n}$ is increasing.
Next, we show that $\sequence {x_n}$ is bounded above.
Using the Binomial Theorem:
\(\ds \paren {1 + \frac 1 n}^n\) | \(=\) | \(\ds 1 + n \paren {\frac 1 n} + \frac {n \paren {n - 1} } 2 \paren {\frac 1 n}^2 + \cdots + \paren {\frac 1 n}^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 1 + \paren {1 - \frac 1 n} \frac 1 {2!} + \paren {1 - \frac 1 n} \paren {1 - \frac 2 n} \frac 1 {3!} + \cdots + \paren {1 - \frac 1 n} \paren {1 - \frac 2 n} \cdots \paren {1 - \frac {n - 1} n} \frac 1 {n!}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds 1 + 1 + \frac 1 {2!} + \frac 1 {3!} + \cdots + \frac 1 {n!}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds 1 + 1 + \frac 1 2 + \frac 1 {2^2} + \cdots + \frac 1 {2^n}\) | (because $2^{n - 1} \le n!$) | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 + \frac {1 - \paren {\frac 1 2}^n} {1 - \frac 1 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 2 \paren {1 - \paren {\frac 1 2}^n}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds 3\) |
So $\sequence {x_n}$ is bounded above by $3$.
From the Monotone Convergence Theorem (Real Analysis), it follows that $\sequence {x_n}$ converges to a limit.
Also see
Note that, although we have proved that this sequence converges to some limit less than $3$ (and incidentally greater than $2$), we have not at this stage determined exactly what this number actually is.
See Euler's number, where this sequence provides a definition of that number (one of several that are often used).
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.19$: Example