Exercise · ZW-UNIFORM-CONVERGENCE-010 · ★★★ Advanced
Let \(f_n(x)=nxe^{-nx}\) on \([0,1]\), with pointwise limit \(0\). Does it converge uniformly to \(0\)?
Choose one option.
Show answer
- B. No, because \(f_n(1/n)=e^{-1}\) does not tend to \(0\).
Explanation
At \(x=1/n\) we have \(f_n(1/n)=e^{-1}\) for every \(n\), so \(\sup|f_n|\ge e^{-1}\) and it does not tend to \(0\). Pointwise convergence is not enough: the peak slides toward \(0\) without shrinking.
Review the concept: Uniform convergence
A sequence of functions \(f_n\) converges uniformly to \(f\) on a set \(A\) if, for every \(\varepsilon>0\), a single index \(N\) works for all points of \(A\).
\[\forall\varepsilon>0\ \exists N:\ n\ge N\Rightarrow|f_n(x)-f(x)|<\varepsilon\ \ \forall x\in A\]
More practice problems: uniform convergence
- ZW-UNIFORM-CONVERGENCE-001 Compute sup_x∈[0,1]|(x)/(n)-0|.
- ZW-UNIFORM-CONVERGENCE-002 Let f_n(x)=x^n on [0,1]. If f is its pointwise limit, what is f(1)?
- ZW-UNIFORM-CONVERGENCE-003 Compute sup_x∈[0,1)|x^n-0|.
- ZW-UNIFORM-CONVERGENCE-004 On which of these sets does x^n converge uniformly to 0?
- ZW-UNIFORM-CONVERGENCE-005 Let f_n(x)=(sin (nx))/(n) on mathbb R. What happens?
- ZW-UNIFORM-CONVERGENCE-006 The f_n are continuous on A and f_n→ f uniformly on A. Then f…
- ZW-UNIFORM-CONVERGENCE-007 Why does x^n not converge uniformly on [0,1]?
- ZW-UNIFORM-CONVERGENCE-008 Compute sup_x≥0(x)/(1+nx).