Test for pointwise and uniform convergence
Test for pointwise and uniform convergence of $u_n(x)=e^{n(\ln x-1)}$ on $(1,e)$.
Explanation of Method
Simplifying $u_n(x)=e^{n\ln x-n}=(x/e)^n$ turns this into a geometric-type sequence with ratio $x/e\in(1/e,1)$ throughout $(1,e)$. As $x\to e^-$, this ratio approaches $1$, so — exactly as in Problem 176 — the pointwise limit $0$ is approached arbitrarily slowly near the right endpoint, blocking uniform convergence on the open interval but not on any sub-interval $[1,b]$ with $b<e$.
- Simplify $u_n(x)=(x/e)^n$.
- Find the pointwise limit and show $\sup_{(1,e)}|u_n|=1$ for every $n$.
- Show uniform convergence holds on $[1,b]$ for $b<e$.
Step 1 · Simplify
$u_n(x)=e^{n\ln x}\cdot e^{-n}=x^n/e^n=(x/e)^n$.
Step 2 · Pointwise limit and failure of uniform convergence
For $x\in(1,e)$: $1/e<x/e<1$, so $(x/e)^n\to0$; the pointwise limit is $u(x)\equiv0$. But $(x/e)^n$ is increasing in $x$ on $(1,e)$, and as $x\to e^-$, $(x/e)^n\to1^n=1$, so $\sup_{x\in(1,e)}(x/e)^n=1$ for every $n$ (approached, not attained). So $\lim_n\sup|u_n-0|=1\ne0$: not uniform.
Step 3 · Uniform convergence on $[1,b]$, $b<e$
Since $(x/e)^n$ is increasing in $x$, its supremum on $[1,b]$ occurs at $x=b$: $\sup_{[1,b]}(x/e)^n=(b/e)^n\to0$ (since $b/e<1$). Uniform convergence holds.
Summary Table
| Interval | Convergence |
|---|---|
| $(1,e)$ | Pointwise only |
| $[1,b]$, $b<e$ | Uniform, rate $(b/e)^n$ |
Additional Notes
- This problem is essentially a relabeled copy of Problem 176: the substitution $y=x/e$ maps $(1,e)$ onto $(1/e,1)$, and the failure of uniform convergence traces to the identical mechanism (the ratio approaching $1$ at the excluded boundary point).
- The connection to Problem 168's logarithmic convergence-region problem is notable: both involve the substitution $y=\ln x$, but here it enters through the exponent $n(\ln x-1)$ rather than as the base of a power series, producing a sequence-of-functions problem instead of a series-convergence-region problem.
- As in Problem 182's close relative Problem 176, the discontinuity or near-discontinuity at the excluded endpoint is entirely a boundary phenomenon: any interval bounded away from $x=e$ recovers full uniform convergence, this time at an explicit geometric rate $(b/e)^n$.
Result
Pointwise on $(1,e)$; not uniform there; uniform on $[1,b]$ for $b<e$: