Consider the following sequences of functions and their integrals. Check whether the pointwise limit function is integrable and whether you can change limit operation and integration:
(i) fn(x) = nx exp (−nx) on [0, 10]
(ii) f (x)= xn /(1+x)2 on [0,1]
(iii) fn(x) = n on [1/k, 2/k] and fn(x) = 0 elsewhere on [0, 2].