Generalized Exponential Limit

Theorem

Let $x \in \R$ be a real number.

Let $\sequence {\nu_n}$ be an increasing, unbounded above real sequence.

Let $\sequence {\xi_n}$ be a real sequence such that:

$\ds \lim_{n \mathop \to \infty} \nu_n \xi_n = x$

Then:

$\ds \lim_{n \mathop \to \infty} \paren {1 + \xi_n}^{\nu_n} = \map \exp x$

where $\exp$ denotes the real exponential function defined as a power series.

Proof

First, let us consider the case where all $\nu_n \in \Z_{\ge 0}$.

Then, by the Binomial Theorem for Integral Index:

$\ds \paren 1 \quad \paren {1 + \xi_n}^{\nu_n} = \sum_{r \mathop = 0}^{\nu_n} \binom {\nu_n} r \xi_n^r$

Now, by Convergent Real Sequence is Bounded, there is some $X \in \R$ such that:

$\size {\nu_j \xi_j} \le X$

for all $j \in \N$.

We will apply Tannery's Theorem, using:

$p_n = \nu_n$

$\map {v_r} n = \dbinom {\nu_n} r \xi_n^r$

$w_r = \dfrac {x^r} {r!}$

$M_r = \dfrac {X^r} {r!}$

For $r \le p_n = \nu_n$, we have that:

$\paren 2 \quad \map {v_r} n = \dfrac {\dbinom {\nu_n} r} {\nu_n^r} \paren {\nu_n \xi_n}^r$

and by Binomial Coefficient over Power Not Greater than Reciprocal of Factorial:

$\dfrac {\dbinom {\nu_n} r} {\nu_n^r} \le \dfrac 1 {r!}$

from which it immediately follows that:

$\size {\map {v_r} n} = \dfrac {\dbinom {\nu_n} r} {\nu_n^r} \size {\nu_n \xi_n}^r \le M_r$

By Radius of Convergence of Power Series over Factorial, we see that:

$\sum_{r \mathop = 0}^\infty M_r$ is convergent

Finally:

Since the conditions are satisfied, we conclude that:

$\Box$

Now, we move to the general case for $\nu_n$.

Without loss of generality, assume that $\nu_n \ge 0$.

First, we must observe that $\xi_n \to 0$ as $n \to \infty$.

This follows by the Squeeze Theorem from:

$0 \le \size {\xi_n} \le \nu_n \size {\xi_n}$

which holds for $n$ such that $\nu_n \ge 1$.

Define two sequences $\sequence {\underline \nu_n}, \sequence {\overline \nu_n}$ as:

$\underline \nu_n = \floor {\nu_n}$

$\overline \nu_n = \floor {\nu_n} + 1$

Then, we have:

$\nu_n - 1 \le \underline \nu_n \le \overline \nu_n \le \nu_n + 1$

and so:

$\nu_n \xi_n - \size {\xi_n} \le \underline \nu_n \xi_n, \overline \nu_n \xi_n \le \nu_n \xi_n + \size {\xi_n}$

hence by the Squeeze Theorem:

$\underline \nu_n \xi_n \to x$

$\overline \nu_n \xi_n \to x$

By the special case already determined, we thus have:

$\paren {1 + \xi_n}^{\underline \nu_n} \to \map \exp x$

$\paren {1 + \xi_n}^{\overline \nu_n} \to \map \exp x$

But, by Power Function is Monotone over Reals:

$\map \min {\paren {1 + \xi_n}^{\underline \nu_n}, \paren {1 + \xi_n}^{\overline \nu_n}} \le \paren {1 + \xi_n}^{\nu_n} \le \map \max {\paren {1 + \xi_n}^{\underline \nu_n}, \paren {1 + \xi_n}^{\overline \nu_n}}$

Hence, once more by the Squeeze Theorem:

$\ds \lim_{n \mathop \to \infty} \paren {1 + \xi_n}^{\nu_n} = \map \exp x$

$\blacksquare$

Sources

• 1926: T.J.I'A. Bromwich: An Introduction to the Theory of Infinite Series (2nd ed.): Art. 57