1. Find the largest subset of R on which the given sequence converges pointwise, and determine the intervals on which the convergence is uniform.
), where fn is defined from part (1a)
2. Let fn be continuous on [a,b] for each n and let fn converge uniformly on (a,b) ∩Q.
Prove that fn converges uniformly on [a,b].
3. Suppose that fn → f and gn → g uniformly on R and f and g are bounded. Prove that fngn → fg uniformly on R.
4. Find the closed form for the following sum
and state its interval of convergence.
5. Find
.
6. (a) State Dirichlet’s Test for Uniform Convergence.
(b) Determine if the following sum converges pointwise, uniformly, or absolutely for all x ∈R,
7. Show that
is integrable on [0,1].
8. Abbott problem #7.2.4
9. Suppose a < 0 and b > 0. Show from the definition, that the function f(x) = x2 is integrable over [a,b] without using the uniform continuity of f.
1
