In this problem we show that pointwise convergence of a
Chapter 11, Problem 4(choose chapter or problem)
In this problem we show that pointwise convergence of a sequence Sn(x) does not implymean convergence, and conversely.(a) Let Sn(x) = nxenx2/2, 0 x 1. Show that Sn(x) 0 as n for each x in0 x 1. Show also thatRn =10[0 Sn(x)]2 dx = n2(1 en)and hence Rn as n . Thus pointwise convergence does not imply mean convergence.(b) Let Sn(x) = xn for 0 x 1, and let f(x) = 0 for 0 x 1. Show thatRn =10[f(x) Sn(x)]2 dx = 12n + 1,and hence Sn(x) converges to f(x) in the mean. Also show that Sn(x) does not converge tof(x) pointwise throughout 0 x 1. Thus mean convergence does not imply pointwiseconvergence.
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