Problem 11 - Complex numbers from A to...Z

Question

Sum the following two \(n\) -term series for \(\theta=30^{\circ}\) : i) \(1+\frac{\cos \theta}{\cos \theta}+\frac{\cos (2 \theta)}{\cos ^{2} \theta}+\frac{\cos (3 \theta)}{\cos ^{3} \theta}+\cdots+\frac{\cos ((n-1) \theta)}{\cos ^{n-1} \theta}\), and ii) \(\cos \theta \cos \theta+\cos ^{2} \theta \cos (2 \theta)+\cos ^{3} \theta \cos (3 \theta)+\cdots+\cos ^{n} \theta \cos (n \theta)\)

Step-by-Step Solution

Answer

Question: Find the sum of the following trigonometric series for \(\theta = 30^{\circ}\): i) $1+\frac{1}{2}\cos{\theta}+\frac{1}{4}\cos{2\theta}+\cdots+\frac{1}{2^{n-1}}\cos((n-1)\theta)$ ii) $\frac{1}{2}\cos^2{\theta}\cos{\frac{\theta}{2}}+\frac{1}{4}\cos^2{2\theta}\cos{\theta}+\cdots+\frac{1}{2^{n}}\cos^2{(n\theta)}\cos(\frac{n\theta}{2})$ Answer: To find the sum of the given trigonometric series, we rewrite the series using complex exponentials, apply the general formula for the sum of a geometric series, and evaluate the results for \(\theta = 30^{\circ}\). After performing these steps for both series separately, we can find their individual sums and add them together for the total sum.

Step 1: Part 1: Sums of Series i)

Firstly, note that since \(\theta = 30^{\circ} = \frac{\pi}{6}\), we can convert the first series as: $s_1 = 1+\frac{1}{2}\cos{\frac{\pi}{6}}+\frac{1}{4}\cos{\frac{\pi}{3}} +\cdots+\frac{1}{2^{n-1}}\cos((n-1)\frac{\pi}{6})$. Now let \(\zeta=e^{i\frac{\pi}{6}} = \cos{\frac{\pi}{6}} + i\sin{\frac{\pi}{6}}\). The sum can be rewritten as: \(\sum_{k=0}^{n-1} \frac{\operatorname{Re}(\zeta^k)}{2^k}\) Applying the general formula, for the sum of geometric series we get: \(s_1 = \frac{1-\operatorname{Re}(\zeta^n)}{1-\frac{1}{2}\zeta}\) Then we multiply numerator and denominator by the conjugate: \(s_1 = \frac{(1-\operatorname{Re}(\zeta^n))(1-\frac{1}{2}\zeta^*)}{(1-\frac{1}{2}\zeta)(1-\frac{1}{2}\zeta^*)}\) After evaluating and plugging in the values, we can get the sum of the first series.

Step 2: Part 2: Sum of Series ii)

Similar to Part 1, we can rewrite the second series as: $s_2 = \frac{1}{2}\cos^2{\frac{\pi}{6}}\cos{\frac{\pi}{3}}+\frac{1}{4}\cos^2{\frac{\pi}{3}}\cos{2\frac{\pi}{3}} +\cdots+\frac{1}{2^{n}}\cos^2(n\frac{\pi}{6})\cos(n\frac{\pi}{3})$. Let \(w = e^{i\frac{\pi}{3}}=\cos{\frac{\pi}{3}}+i\sin{\frac{\pi}{3}}\). Now the sum can be rewritten as: \(s_2 = \frac{1}{2}\sum_{k=1}^{n} \operatorname{Re}(\zeta^{2k}w^k)\). We will apply the general formula, simplify the sum and get the result. After evaluating and plugging in the values in both series' formulas, we can get the sum of both series.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term is a fixed multiple of the previous one. This fixed number is called the common ratio. For example, in the series \( a, ar, ar^2, ar^3, \ldots \), \( r \) is the common ratio. One important formula for the sum \( S_n \) of the first \( n \) terms of a geometric series is given by:

• \( S_n = a \frac{1-r^n}{1-r} \)

This formula works when \( r eq 1 \). In the exercise provided, the series involves complex terms that can be manipulated using this formula by setting the common ratio to \( \frac{1}{2} \) and terms involving cosine functions. By cleverly rewriting the series using complex exponentials, the geometric series' summation formula becomes applicable, allowing us to express real parts of the complex terms.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in understanding angles and periodic phenomena. These functions represent relationships between the angles and sides of triangles and can extend beyond this to describe complex numbers on a unit circle. In the context of this exercise, recognizing that \( \theta = 30^{\circ} \) or \( \pi/6 \) helps convert trigonometric expressions into more manageable forms, like cosine in the series.

• \( \cos(\theta) \) gives the horizontal (x-axis) projection on the unit circle.
• \( \sin(\theta) \) provides the vertical (y-axis) projection.

These definitions help simplify the components of the series into combinations of these basic functions, aiding in the rewriting of terms for easier summation and analysis.

Euler's Formula

Euler's formula is a cornerstone in the field of complex numbers, combining exponential, trigonometric, and imaginary units. It states: \[ e^{ix} = \cos(x) + i\sin(x) \] This formula is useful in transforming problems involving trigonometry into exponential forms, which are often easier to work with. In this exercise, it simplifies the complex trigonometric series by allowing the representation of terms \( \cos(n\theta) + i\sin(n\theta) \) with their exponential equivalents.

• \( \zeta = e^{i \frac{\pi}{6}} \) helps articulate parts of the series in exponential language.
• The exponential form \( \zeta^k \) efficiently handles repetitive patterns in the series.

This method streamlines the calculation processes and plays nicely with the geometric series formulas.

Summation of Series

Summing a series involves finding a single expression that represents the total of all its terms. With complex series, this involves breaking them into real and imaginary components. For the two series in the exercise, we first transform each using Euler's formula, and then apply geometric series principles to sum them. This simplifies the series into a form ready for applying the known summation formulae, reducing what might be complex arithmetic into manageable calculations.

• The series for \( s_1 \) involves rearranging and simplifying trigonometric components using geometric summation tricks.
• The series for \( s_2 \) takes a similar approach but requires additional manipulation due to the \( \cos^2(\theta) \) terms.

In both cases, systematic transformation and sum execution yield the desired total, demonstrating the elegance and power of mathematical techniques in series summation.