. For n = 1, 2, 3 . . ., let fn be a Riemann integrable function defined on the interval [n, n + 1]. Define a function f : [1, ) R by f(x) = fn(x) when n x < n + 1.

If R n+1 n |fn| = 1 n3 for all n, prove that the improper integral R 1 f is absolutely convergent.

(b) Let an = R n+1 n fn. Give an example of functions fn for which X n=1 an converges but R 1 f diverges. Hint: consider a periodic function.

(c) For each n, extend the definition of fn to R by setting fn(x) = 0 for x 6 [n, n + 1]. Prove that fn 0 pointwise on R by showing that for every x R the sequence {fn(x)} is equal to zero for all sufficiently large n N.

(d) If fn(x) = 1 for x [n, n + 1], does fn converge to zero uniformly on R? What about if fn = 1 n for x [n, n + 1]? Prove your answers. As in part (c), fn(x) = 0 for x not belonging to [n, n + 1].