Prove that Z b a ∞ X n=1 fn(x

1. Quizz 12 (1) Let (f_n) be a sequence in C[a, b] such thatP∞

n=1f_nis convergent in (C[a, b], k · k∞).

Prove that

Z b a

∞

X

n=1

fn(x)

! dx =

∞

X

n=1

Z b a

fn(x)dx.

(2) Evaluate

∞

X

n=0

1

(n + 1)2ⁿ using the series of functions

∞

X

n=0

xⁿ

2ⁿ, x ∈ [0, 1].

(3) Let (f_n) be a sequence of real valued functions on [a, b]. Suppose that (a) For each x0 ∈ [a, b],

f (x0) = lim

n→∞fn(x0)

exist. In other words, f defines a real valued function on [a, b].

(b) f_n⁰ : [a, b] → R is continuous on [a, b] such that the sequence (fn⁰) is convergent to g in (C[a, b], k · k∞). Here g ∈ C[a, b].

Prove that the function f : [a, b] → R is C¹ such that f⁰(x) = g. Hint:

f_n(x) = f_n(a) + Z x

a

f_n⁰(t)dt.

1