Test for pointwise and uniform convergence of u_n(x)=(1+
Test for pointwise and uniform convergence of $u_n(x)=\left(1+\dfrac xn\right)^n$: (1) on a bounded interval $(a,b)$; (2) on $(-\infty,\infty)$.
Explanation of Method
This is the classical limit defining $e^x$. Expanding $n\ln(1+x/n)$ to second order gives an explicit error term of size $\sim x^2e^x/(2n)$, which is uniformly bounded on any bounded interval (giving uniform convergence there) but grows without bound as $|x|\to\infty$ for each fixed $n$ (ruling out uniform convergence on all of $\mathbb R$).
- Establish the pointwise limit $u(x)=e^x$.
- Bound the error on a bounded interval $(a,b)$ via Taylor expansion.
- Show the error is unbounded on all of $\mathbb R$ for each fixed $n$.
Step 1 · Pointwise limit
$n\ln(1+x/n)=n\left[\frac xn-\frac{x^2}{2n^2}+O(n^{-3})\right]=x-\frac{x^2}{2n}+O(n^{-2})\to x$, so $u_n(x)=e^{n\ln(1+x/n)}\to e^x$ for every fixed $x$.
Step 2 · Uniform convergence on $(a,b)$
From the expansion: $u_n(x)=e^x\cdot e^{-x^2/(2n)+O(1/n^2)}=e^x\left(1-\frac{x^2}{2n}+O(n^{-2})\right)$, so $$ |u_n(x)-e^x| \lesssim \frac{x^2e^x}{2n} \le \frac{M^2e^M}{2n}, \qquad M=\max(|a|,|b|), $$ a bound independent of $x\in(a,b)$ that $\to0$. Uniform convergence holds on every bounded interval.
Step 3 · Failure on $(-\infty,\infty)$
For each fixed $n$, as $x\to+\infty$: $|u_n(x)-e^x|\sim x^2e^x/(2n)\to\infty$. So $\sup_{x\in\mathbb R}|u_n(x)-e^x|=\infty$ for every $n$ — uniform convergence fails on the full real line.
Summary Table
| Domain | Convergence |
|---|---|
| $(a,b)$ (bounded) | Uniform |
| $(-\infty,\infty)$ | Pointwise only |
Additional Notes
- This is a striking example where the failure of uniform convergence has nothing to do with a boundary point of a fixed interval (as in Problems 176, 181, 182) but rather with the sheer unboundedness of the domain itself: the error grows because $e^x$ itself grows without bound.
- The rate $O(x^2e^x/n)$ derived here from Taylor expansion is the same type of second-order correction seen in Problem 134's perturbed alternating series, both arising from expanding a nonlinear function ($\ln(1+\cdot)$ there and here) to quadratic order in a small parameter.
- This example underlies why term-by-term limit interchange (e.g. differentiating or integrating $(1+x/n)^n$ to recover properties of $e^x$) is only rigorously justified on bounded intervals, unless a stronger uniform bound valid on all of $\mathbb R$ is separately established.
Result
Converges to $e^x$ uniformly on every bounded interval, but not uniformly on $\mathbb R$: