Prove that n→∞lim Z b a fn(x)dx = Z b a f (x)dx

1. Quizz 11

(1) Let (f_n) be a sequence in C[a, b] such that (f_n) is convergent to f in (C[a, b], k · k∞).

Prove that

n→∞lim Z _b

a

f_n(x)dx = Z _b

a

f (x)dx.

(2) Let fn(x) = nx(1 − x²)ⁿ for x ∈ [0, 1].

(a) Let x₀ ∈ [0, 1]. Prove that

n→∞lim f_n(x₀) = 0.

(b) Is (f_n) convergent to the zero function in (C[0, 1], k · k∞)? (Hint: consider (1)).

(3) Let (f_n) be a sequence in (C[a, b], k · k∞). Suppose that (f_n) is convergent to f in (C[a, b], k · k∞). Prove or disprove that (f_n²) is convergent to f² in (C[a, b], k · k∞).

(4) Let (a_n) be a sequence of real numbers such thatP∞

n=1|a_n| is convergent. Define f (x) =

∞

X

n=1

ansin nx, x ∈ [0, 2π].

(a) Show that f defines a real valued continuous function on [0, 2π], i.e. f ∈ C[0, 2π].

(b) Let k be a natural number. Find Z 2π

0

f (x) sin kxdx in terms of an. (c) Find

Z 2π 0

|f (x)|²dx in terms of an.

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