170 CHAPTER 6. INFINITE SERIES

9. Suppose { fn} is a sequence of real valued functions which converges uniformly to acontinuous function f . Can it be concluded the functions fn are continuous? Explain.

10. Let h(x) be a bounded continuous function. Show the function f (x) = ∑∞n=1

h(nx)n2 is

continuous.

11. Let S be a any countable subset of R. This means S is actually the set of terms of asequence. That is S = {sn}∞

n=1. Show there exists a function f defined on R which isdiscontinuous at every point of S but continuous everywhere else. Hint: This is realeasy if you do the right thing. It involves Theorem 6.8.4 and the Weierstrass M test.

12. By Theorem 4.10.3 there exists a sequence of polynomials converging uniformly tof (x) = |x| on the interval [−1,1] . Show there exists a sequence of polynomials, {pn}converging uniformly to f on [−1,1] which has the additional property that for alln, pn (0) = 0.

13. If f is any continuous function defined on [a,b] , show there exists a series of the form∑

∞k=1 pk, where each pk is a polynomial, which converges uniformly to f on [a,b].

Hint: You should use the Weierstrass approximation theorem to obtain a sequenceof polynomials. Then arrange it so the limit of this sequence is an infinite sum.

14. Sometimes a series may converge uniformly without the Weierstrass M test beingapplicable. Show ∑

∞n=1 (−1)n x2+n

n2 converges uniformly on [0,1] but does not con-verge absolutely for any x ∈R. To do this, it might help to use the partial summationformula, 6.7. Note that ∑

∞n=1 (−1)n x2+n

n2 = ∑∞n=1 (−1)n x2+n

n

( 1n

).