MAT 542  Assignment No. 2  Due: Feb. 15, 2002

(1) Under the hypothesis that {a_{n}} and {b_{n}} are positive sequences, prove that: a)

\overline{lim}_{n}a_{n}b_{n} ≤ \overline{lim}_{n}a_{n}\overline{lim}_{n}b_{n},

provided the right side is not the indeterminate form 0×∞. Show by example that strict inequality may hold.

b) If lim_{n} a_{n} exists, then the equality holds in a) provided the right side is not indeterminate; that is, show that inn this case,

\overline{lim}_{n}a_{n}b_{n} = lim_{n} a_{n}\overline{lim}_{n}b_{n}.

(2) Find all the roots of cos z = 2.

(3) a) Find: ℜ(sin z) , ℑ(sin z) , ℜ(cos z) , ℑ(cos z) .

b) Prove that

| sin z|^{2} = sin^{2} x + sinh^{2} y (z = x + ıy) , | cos z|^{2} = cos^{2} x + sinh^{2} y ,

where cosh z = \frac{e^{z}+e^{−z}}{2} and sinh z = \frac{e^{z}−e^{−z}}{2} .

c) Derive the addition formula for cosh(a+b) and sinh(a+b).

d) Evaluate D sinh z, D cosh z, and cosh^{2} z − sinh^{2} z.

(4) Prove, using power series, that e^{−z} = \frac{1}{e^{z}} .

(5) Is it always true that Log(e^{z}) = z? Support your answer with either a proof or a counter-example.

(6) Let {k_{n}} be a sequence in which every positive integer appears once and only once. Let ∑ a_{n} be a series, putting a′_{n} = a_{k_{n}},we say that ∑ a′_{n} is a \underline{rearrangement} of ∑ a_{n}.

a) Let a_{n} ∈ ℝ. Assume that ∑ a_{n} converges but ∑ |a_{n}| does not. Let a ∈ ℝ. Show that there is a rearrangement ∑ a′_{n} of ∑ a_{n} such that a = ∑ a′_{n}.

b) Show that ∑ a_{n} converges absolutely if and only if every

rearrangement converges to the same sum.

(7) Let {a_{n}} be a real sequence. Show that

\overline{lim}_{n} \{a_{n}\} = sup \{α; α = lim_{n} b_{n}\} ,

with{b_{n}} a convergent subsequence of a_{n}

\underline{lim}_{n}a_{n} = inf \{α; α = lim_{n} b_{n} \} ,

with {b_{n}} as above. In this exercise a sequence {b_{n}} with lim b_{n} = +∞ (similarly −∞) is to be considered a convergent sequence.

(8) (Only for those who know some algebra.) Show that ℂ is the only non-trivial finite dimensional commutative division algebra over ℝ.

(9) a) Given two points z_{1}, z_{2}, |z_{1}| < 1, |z_{2}| < 1, show that for every point z ≠ 1 in the closed triangle with vertices z_{1}, z_{2} and 1,

\frac{|1 − z|}{1 − |z|} ≤ K,

where K is a constant that depends only on z_{1} and z_{2}.

b) Determine the smallest value of K for z_{1} = \frac{1+ı}{2} and z_{2} = \frac{1−ı}{2}.