Find an example of a sequence of continuous functions (fn) on (0, l) that conver

Question

Find an example of a sequence of continuous functions (fn) on (0, l) that converges pointwise to 0, but integral 0 1 fn rightarrow + infinity. Justify your assertions. (b) Find a sequence of continuous functions (gn) that converges pointwise but not uniformly to 0 on [0,1] such that. integral 0 1 gn rightarrow 0. (Hence lack of uniform convergence does not necessarily mean the limit of the integrals behaves badly)

Explanation / Answer

1) (Not entirely sure about this part) consider fn(x) = (1+1/2+1/3 ... 1/n)*[x(1-x)]^n fn(x) converges pointwise to 0 But S fn(x) -> to infinity 2) (sure about this) gn(x) = n^2e^(-nx) This converges pointwise to 0 But S gn(x) -> 0 Hence uniform converge is not a necessary condition for good behaviour of limit of integration of functions. For more info on Part II, read this: http://dana.ucc.nau.edu/imf2/M414notes24.pdf

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.