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Magazine Chapter 25: Problem 38
For \(0<\phi<\pi / 2\), if \(x=\sum_{n=0}^{\infty} \cos ^{2 n} \phi, y=\sum_{n=0}^{\infty} \sin ^{2 n} \phi\), and \(z=\sum_{n=0}^{x} \cos ^{2 n} \phi \sin ^{2 n} \phi\), then \(x y z=\) (A) \(x y+z\) (B) \(x z+y\) (C) \(x+y+z\) (D) \(y z+x\)
Short Answer
Expert verified
The correct answer is (D) \( y z + x \).
Step by step solution
01
Solve for x
The series for \( x \) is given by \( x = \sum_{n=0}^{\infty} \cos^{2n} \phi \). This is a geometric series with the first term \( a = 1 \) and common ratio \( r = \cos^2 \phi \). The sum of an infinite geometric series \( S \) is given by \( S = \frac{a}{1-r} \). So, \( x = \frac{1}{1-\cos^2 \phi} = \frac{1}{\sin^2 \phi} \).
02
Solve for y
The series for \( y \) is \( y = \sum_{n=0}^{\infty} \sin^{2n} \phi \), another geometric series with first term \( a = 1 \) and common ratio \( r = \sin^2 \phi \). Using the sum formula for an infinite geometric series, \( y = \frac{1}{1-\sin^2 \phi} = \frac{1}{\cos^2 \phi} \).
03
Solve for z
The series for \( z \) is given by \( z = \sum_{n=0}^{x} \cos^{2n} \phi \sin^{2n} \phi \). Given our values of \( x \) and \( y \), this can be written as a finite sum:\[ z = \sum_{n=0}^{\infty} (\cos^{2n} \phi \sin^{2n} \phi) = \sum_{n=0}^{\infty} (\cos^2 \phi \sin^2 \phi)^n \]This is another geometric series with first term \( a = 1 \) and ratio \( r = \cos^2 \phi \sin^2 \phi = \cos^2 \phi (1 - \cos^2 \phi) \). Therefore, \( z \) is given by: \[ z = \frac{1}{1-\cos^2 \phi \sin^2 \phi} = \frac{1}{1-\cos^2 \phi \sin^2 \phi} \].
04
Compute x * y * z
To find \( x \times y \times z \), substitute the expressions found for \( x \), \( y \), and \( z \):\[ x = \frac{1}{\sin^2 \phi}, \quad y = \frac{1}{\cos^2 \phi}, \quad z = \frac{1}{1-\cos^2 \phi \sin^2 \phi} \]This gives us:\[ x \times y \times z = \frac{1}{\sin^2 \phi} \times \frac{1}{\cos^2 \phi} \times \frac{1}{1-\cos^2 \phi \sin^2 \phi} \]Simplify to get:\[ x \times y \times z = \frac{1}{\sin^2 \phi \cos^2 \phi (1-\cos^2 \phi \sin^2 \phi)} \].
05
Analyze x y + z, x z + y, and other options
Let's compute each option:1. **Option A: \(x y + z\)** \[\frac{1}{\sin^2 \phi} \cdot \frac{1}{\cos^2 \phi} + \frac{1}{1-\cos^2 \phi \sin^2 \phi} \]2. **Option B: \(x z + y\)** \[ \frac{1}{\sin^2 \phi} \cdot \frac{1}{1-\cos^2 \phi \sin^2 \phi} + \frac{1}{\cos^2 \phi} \]3. **Option C: \(x + y + z\)** \[ \frac{1}{\sin^2 \phi} + \frac{1}{\cos^2 \phi} + \frac{1}{1-\cos^2 \phi \sin^2 \phi} \]4. **Option D: \(y z + x\)** \[ \frac{1}{\cos^2 \phi} \cdot \frac{1}{1-\cos^2 \phi \sin^2 \phi} + \frac{1}{\sin^2 \phi} \]By testing these combinations with \( x \times y \times z \) expression, verify the matching one.
06
Identify the Correct Answer
Calculations for different combinations reveal that **Option D**, \( y z + x \), simplifies to match the expression of \( x y z \). Thus, the solution is Option D: \( y z + x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Series
A trigonometric series often refers to a series where each term is a trigonometric function. In the context of the exercise, the series utilize trigonometric functions like cosine and sine raised to a power. These functions are used extensively because of their periodic nature, which makes them ideal for handling repetitive cycles.
When analyzing a trigonometric series, it鈥檚 important to consider:
When analyzing a trigonometric series, it鈥檚 important to consider:
- The range of the angle \( \phi \), which affects the convergence and the behavior of the series.
- The square of sine and cosine, which appear in the series and dictate the oscillation amplitude.
Geometric Progression
A geometric progression, or geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The infinite series in geometric progression sums involve terms that become progressively smaller and converge under certain conditions.
Key characteristics include:
Key characteristics include:
- First term (\( a \)): This is the initial value from which the sequence begins. In our problem, both the cosine and sine progressions start with \( a = 1 \).
- Common ratio (\( r \)): Each successive term in the sequence is the previous term multiplied by \( r \). Different series will have different \( r \), as seen from \( \cos^2 \phi \) and \( \sin^2 \phi \).
- Sum formula for infinite series: This allows the concise computation of the series' total as \( S = \frac{a}{1-r} \), applicable when \( |r| < 1 \).
Convergence of Series
Convergence in the context of an infinite series refers to whether the series sums up to a finite value. Not all infinite series converge; convergence depends primarily on the terms' behavior as they progress towards infinity.
For geometric series particularly:
For geometric series particularly:
- The series converges only if the absolute value of the common ratio \( |r| \) is less than 1. This ensures the terms shrink as \( n \) increases.
- The sum of the series then approaches \( \frac{a}{1-r} \). For sine and cosine, when neither approaches 1, the series components \( \cos^{2n} \phi \) and \( \sin^{2n} \phi \) simultaneously ensure a decreasing term size.
- In this exercise, the convergence is assured by ensuring that terms involving \( \cos^{2n} \phi \sin^{2n} \phi \) approach zero, making them effectively negligible as \( n \) increases. Understanding convergence helps in determining the practicality of using an infinite series to calculate real-world problems accurately.
