Complex Analysis, Spring 2011.

Instructor: Dmitry Ryabogin

Assignment IV.

1. Problem 1. a) Expand (1 − z)^{−m} m a positive integer, in powers of z.

b) Expand \frac{2z+3}{z+1} in powers of z − 1. What is the radius of convergence?

2. Problem 2. a) If ∑ a_{n}z^{n} has radius of convergence R, what is the radius of convergence of ∑ a_{n}z^{2n}? of ∑ a_{n}^{2}z^{n}?

b) If f(z) = ∑ a_{n}z^{n}, what is ∑ n^{3}a_{n}z^{n}?

c) If ∑ a_{n}z^{n} and ∑ b_{n}z^{n} have radii of convergence R_{1} and R_{2}, show that the radius of convergence of ∑ a_{n}b_{n}z^{n} is at least R_{1}, R_{2}.

d) If lim_{n→∞} |a_{n+1}|/|a_{n}| = R, prove that ∑ a_{n}z^{n} has radius of convergence R. Hint: Consider

|a_{n}| = \frac{|a_{n}|}{|a_{n−1}|} \frac{|a_{n−1}|}{|a_{n−2}|} ... \frac{|a_{k+1}|}{|a_{k}|}|a_{k}|

to show that

lim_{n→∞} |a_{n+1}|/|a_{n}| = R yields  lim_{n→∞} |a_{n}|^{1/n} = R.

3. Problem 3. a) Find the radius of convergence of the following power series:

ℵ) ∑ n^{p}z^{n},  ℶ) ∑ \frac{z^{n}}{n!} ,  ℷ) ∑ q^{n^{2}}z^{n}(|q| < 1),  ℸ) ∑ z^{n!}.

b) Describe the convergence of the series in a) for |z| = 1.

4. Problem 4.

a) For what values of z is

∑_{n=0}^{∞} (\frac{z}{1 + z} )^{n}

convergent?

b) Same question for

∑_{n=0}^{∞} \frac{z^{n}}{1 + z^{2n}} .

Hint: Does there exists any difference between |z| < 1 and |z| > 1?

5. Problem 5*. The purpose of this exercise is to prove a weak version of the following Theorem of Levy and Steinitz: If a series A = z_{1} + z_{2} + z_{3} + ... converges, but not absolutely, then there exists a line l such that for every s ∈ l there exists a permutation σ(n) of indices of A, such that the permuted series A′ = a_{σ(1)}+a_{σ(2)}+a_{σ(3)}+... converges to s.

Definition. Let {r_{n}}, {s_{m}} be two increasing sequences of natural numbers, such that r_{n} < r_{n+1}, s_{m} < s_{m+1}, r_{m} ≠ s_{m} ∀m, n, and ∪_{n,m}(r_{n} ∪ s_{m}) = N. Then two series

A_{r} := z_{r_{1}} + z_{r_{2}} + z_{r_{3}} + ... and A_{s} := z_{s_{1}} + z_{s_{2}} + z_{s_{3}} + ...

are called complemented parts of a series A = z_{1} + z_{2} + z_{3} + .... (In other words, you color the terms of A in red and blue, say).

If a permutation σ, σ : N → N, of terms of A does not change the order inside A_{r}, A_{s} (if z_{r_{m}} appears before z_{r_{n}} and z_{s_{m}} appears before z_{s_{n}} for any m, n, m < n, after permutation), then one says that σ moves complemented parts A_{r}, A_{s} with respect to each other.

Prove a)- f) and obtain the proof.

a) The series A_{s}(A_{r}) is convergent, provided A and A_{r}(A_{s}) are convergent. If a permutation σ moves complemented parts A_{r}, A_{s} with respect to each other, then the sum z_{σ(1)} + z_{σ(2)} + z_{σ(3)} + ... remains unchanged (is A).

b) If a real A is convergent, but not absolutely, and if one of the complemented parts, say, A_{r} diverges to ∞, then another complemented part, A_{s} diverges to −∞.

c) If a real A is convergent, but not absolutely, and if all the terms of one of the complemented parts, say, A_{r}, have the same signs, then, using only the permutations that move the complemented parts, one can obtain any sum for the permuted series.

d) If a series |z_{1}|+|z_{2}|+|z_{3}|+... diverges, then there exists a direction of concentration α, such that ∀ϵ > 0, the series of absolute values of terms that are located in the angle α − ϵ < arg z < α + ϵ is divergent.

e) If A is convergent, but not absolutely, and if a direction of concentration of A is a positive part of the real axes, prove that one can find {r_{n}} such that Rez_{r_{1}} + Rez_{r_{2}} + Rez_{r_{3}} + ... → ∞, but Imz_{r_{1}} + Imz_{r_{2}} + Imz_{r_{3}} + ... is convergent.

f) Prove the Theorem of Levy and Steinitz.