Exercise · ZW-UNIFORM-CONVERGENCE-009 · ★★★ Advanced

Let \(f_n(x)=x^n(1-x)\) on \([0,1]\), with pointwise limit \(0\). Does it converge uniformly?

Choose one option.

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D. Yes, because its maximum is \(\frac1{n+1}\left(\frac n{n+1}\right)^n\le\frac1{n+1}\).

Explanation

The maximum is attained at \(x=\frac n{n+1}\) and equals \(\frac1{n+1}\left(\frac n{n+1}\right)^n\le\frac1{n+1}\to0\). The factor \(1-x\) tames the bad behavior of \(x^n\) near \(1\).

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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 \(A\).

\[\forall\varepsilon>0\ \exists N:\ n\ge N\Rightarrow|f_n(x)-f(x)|<\varepsilon\ \ \forall x\in A\]

Let \(f_n(x)=x^n(1-x)\) on \([0,1]\), with pointwise limit \(0\). Does it converge uniformly?

Explanation

Review the concept: Uniform convergence

More practice problems: uniform convergence