Section 4

Sequences of Complex Numbers

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Problem 3.36

Assume that the numbers z1,z2,…,zn,…z_{1}, z_{2}, \ldots , z_{n}, \ldots are all in the sector −α≤arg⁡z≤α-\alpha \leq \arg z \leq \alpha, α<π2\alpha < \frac{\pi }{2}. Then the series

z1+z2+⋯+zn+⋯and∣z1∣+∣z2∣+⋯+∣zn∣+⋯ z_{1} + z_{2} + \cdots + z_{n} + \cdots \quad \text{and} \quad \left|z_{1}\right| + \left|z_{2}\right| + \cdots + \left|z_{n}\right| + \cdots

are either both convergent or both divergent.

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Problem 3.37

Suppose that the numbers z1,z2,…,zn,…z_{1}, z_{2}, \ldots , z_{n}, \ldots are all in the half-plane Re(z)≥0\mathrm{Re}\left(z\right) \geq 0 and that the two series

z1+z2+⋯+zn+⋯andz12+z22+⋯+zn2+⋯ z_{1} + z_{2} + \cdots + z_{n} + \cdots \quad \text{and} \quad z_{1}^{2} + z_{2}^{2} + \cdots + z_{n}^{2} + \cdots

converge. Then ∣z1∣2+∣z2∣2+⋯+∣zn∣2+⋯\left|z_{1}\right|^{2} + \left|z_{2}\right|^{2} + \cdots + \left|z_{n}\right|^{2} + \cdots converges too.

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Problem 3.38

There exist complex sequences z1,z2,…,zn,…z_{1}, z_{2}, \ldots , z_{n}, \ldots for which all the series

z1k+z2k+⋯+znk+⋯ ,k=1,2,3,… z_{1}^{k} + z_{2}^{k} + \cdots + z_{n}^{k} + \cdots , \quad k = 1, 2, 3, \ldots

converge and all the series

∣z1∣k+∣z2∣k+⋯+∣zn∣k+⋯ ,k=1,2,3,… \left|z_{1}\right|^{k} + \left|z_{2}\right|^{k} + \cdots + \left|z_{n}\right|^{k} + \cdots , \quad k = 1, 2, 3, \ldots

diverge.

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Problem 3.39

Let z1,z2,…,zn,…z_{1}, z_{2}, \ldots , z_{n}, \ldots be arbitrary complex numbers. If there exists a positive distance δ\delta such that ∣zl−zk∣≥δ\left|z_{l} - z_{k}\right| \geq \delta for l<kl < k, l,k=1,2,3,…l, k = 1, 2, 3, \ldots the convergence exponent of the sequence ∣z1∣,∣z2∣,∣z3∣,…\left|z_{1}\right|, \left|z_{2}\right|, \left|z_{3}\right|, \ldots is at most 22. [Problem 114 of Part I.]

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Problem 3.40

The limit points of the complex numbers

1iα+2iα+3iα+⋯+niαn,α real,  α≥0,n=1,2,3,… \frac{1^{i\alpha } + 2^{i\alpha } + 3^{i\alpha } + \cdots + n^{i\alpha }}{n}, \quad \alpha \text{ real}, \; \alpha \geq 0, \quad n = 1, 2, 3, \ldots

fill out the entire circle with radius (1+α2)−1/2(1+\alpha^{2})^{-1/2} and center at the origin.

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Remark.
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The expression in question is closely related to a sum of rectangles.

Problem 3.41

Find the locus of the limit points of the complex sequence z1,z2,…,zn,…z_{1}, z_{2}, \ldots , z_{n}, \ldots, where

zn=(1+i1)(1+i2)(1+i3)⋯(1+in). z_{n} = \left(1 + \frac{i}{1}\right)\left(1 + \frac{i}{2}\right)\left(1 + \frac{i}{3}\right)\cdots \left(1 + \frac{i}{n}\right).
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Problem 3.42

Put

(1+i1)(1+i2)⋯(1+in)=zn \left(1+\frac{i}{\sqrt{1}}\right)\left(1+\frac{i}{\sqrt{2}}\right)\cdots \left(1+\frac{i}{\sqrt{n}}\right) = z_{n}

and connect the points zn−1z_{n-1} and znz_{n} by a straight line. The distance between these two points is always 11. The polygonal line connecting the successive points approaches with increasing nn an Archimedean spiral; if zn=rneiφnz_{n} = r_{n}e^{i\varphi_{n}}, rn>0r_{n} > 0, 0<φn−φn−1<π20 < \varphi_{n} - \varphi_{n-1} < \frac{\pi }{2}, then

lim⁡n→∞rn−rn−1φn−φn−1=12,lim⁡n→∞rnφn=12. \lim _{n\to \infty } \frac{r_{n} - r_{n-1}}{\varphi _{n} - \varphi _{n-1}} = \frac{1}{2}, \quad \lim _{n\to \infty } \frac{r_{n}}{\varphi _{n}} = \frac{1}{2}.
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Problem 3.43

Let tt be a fixed real number and put z=2neitnz = 2ne^{\frac{it}{\sqrt{n}}}. Then

lim⁡n→∞nπ 2zn!z(z−1)(z−2)⋯(z−n)=e−t2. \lim _{n\to \infty } \sqrt{\frac{n}{\pi }}\, \frac{2^{z}n!}{z(z-1)(z-2)\cdots (z-n)} = e^{-t^{2}}.
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Remark.
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Cf.problems 59 and 10 of Part II.