T E C H N I S C H E UNIVERSIT ¨ AT DARMSTADT

Fachbereich Mathematik Prof. Dr. W. Trebels Dr. V. Gregoriades Dr. A. Linshaw

T E C H N I S C H E UNIVERSIT ¨ AT DARMSTADT

A

2nd Tutorial Analysis II (engl.)

Summer Semester 2010

(T2.1)

Let a, bsuch that −∞ ≤a≤b≤+∞.

1. Suppose that f : (a, b) → C is an absolutely integrable function. Prove that the function f is integrable as well. (This is Lemma 4.6 Chap. 5).

2. If f, g : (a, b) :→R are admissible functions such that|f(x)| ≤g(x) for all x∈(a, b) and the function g is integrable, then the function f is absolutely integrable. (This is Theorem 4.7 Chap. 5 - Comparison test for integrals).

(T2.2)

Suppose that f : [1,∞) is a strictly increasing function. Prove that f(1) +. . .+f(n−1)<

Z n

1

f(x)dx < f(2) +. . .+f(n) for all n ≥ 2. Choosing f = log : (0,∞) → R infer that nⁿ

eⁿ⁻¹ < n! < (n+ 1)ⁿ⁺¹

eⁿ for all n≥2 and so lim

n→∞

√n

n!

n = 1

e. (T2.3)

Suppose that f : R→ R is a 2π-periodic function which is jump continuous andeven (i.e., f(x) = f(−x) for x∈R). Prove that the Fourier series of f is

a₀ 2 +

∞

X

n=1

a_n·cos(nx)

for some sequence of real numbers (an)n≥0. What happens iff is odd (i.e, f(−x) =−f(x) for all x∈R)?