Problem 28 - A Complete Resource Book in Mathematics for JEE Main

Question

Range of values of \(f(x)=1+\sin x+\sin ^{3} x+\sin ^{5} x \ldots\) \(x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)\) is (A) \((0,1)\) (B) \((-\infty, \infty)\) (C) \((-2,2)\) (D) None of these

Step-by-Step Solution

Answer

The range of \( f(x) \) is \((-\infty, \infty)\), so it's option (B).

Step 1: Understand the Infinite Series

The given function is an infinite series: \( f(x) = 1 + \sin x + \sin^3 x + \sin^5 x + \ldots \). This is a geometric series with the first term \( a = \sin x \) and the common ratio \( r = (\sin^2 x) \). We need to analyze the sum of this series.

Step 2: Derive the Sum of Infinite Series

To find the sum of the infinite series \( S \), we use the geometric series sum formula: \( S = \frac{a}{1-r} \), provided \(|r| < 1\). Here, \( a = \sin x \) and \( r = \sin^2 x \), so the sum of the series is \( \frac{\sin x}{1 - \sin^2 x} \). This simplifies to \( \frac{\sin x}{\cos^2 x} \) since \(1 - \sin^2 x = \cos^2 x\).

Step 3: Simplify the Sum

Now we simplify the expression further by noting that \( \frac{\sin x}{\cos^2 x} = \tan x \sec x = \sec x \tan x \). Thus, the series sum is \( \sec x \tan x \).

Step 4: Analyze the Function Behavior

The function becomes \( f(x) = 1 + \sec x \tan x \). We know that \( \tan x \) and \( \sec x \) both have vertical asymptotes as \( x \to \frac{\pi}{2}^- \) and \( x \to \frac{-\pi}{2}^+ \). Thus, the function can grow quite large in magnitude. However, within \( x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right) \), both functions adjust such that all values encompassing zero, positive and negative, are possible within a finite range.

Step 5: Determine the Range

Since \( \sec x \tan x \) can take any real value from \(-\infty\) to \( \infty \) as \( x \to \pm \frac{\pi}{2} \), \( f(x) = 1 + \sec x \tan x \) can take any real value as well. Hence, the range of \( f(x) \) is \((-\infty, \infty)\).

Key Concepts

Infinite Series

An infinite series is a sum of infinitely many terms. In mathematics, an infinite series like the geometric series can provide great insights about function behaviors beyond finite boundaries. For the given function, we identified an infinite series:

• The first term, called the initial term, is often denoted as "a." Here, that's "\( \sin x \)."
• The common ratio for our series is "\( \sin^2 x \)."

The beauty of infinite series, particularly geometric series, lies in their convergence properties. A series converges if the absolute value of the common ratio is less than 1. Using the sum formula for infinite geometric series, \[ S = {\frac{a}{1-r}} \],we can simplify our series to the expression: \( { \frac{\sin x}{\cos^2 x} } = \tan x \sec x \).This provides us a clearer view of how the infinite terms affect the overall function behavior.

Range of Functions

Understanding the range of a function is crucial because it tells us all possible outputs of a function given its domain. For a function to be well understood, its behavior across its entire range must be grasped. With the function in question:

• It combines a constant term, "1", with the expression \( \sec x \tan x \).
• This combination results from the sum of the infinite series.

Focusing on the domain \( x \in \left(\frac{-\pi}{2}, \frac{\pi}{2}\right) \), both \( \tan x \) and \( \sec x \) reach vertical asymptotes approaching the vertical boundaries of the interval. This implies that as \(x\) comes close to these boundaries, the function approaches infinity or negative infinity. Hence, the range of our function spanning the real numbers is all real values from \((-\infty, \infty)\).

Trigonometric Functions

Trigonometric functions are fundamental components of mathematics, describing the relationships involving angles and sides of triangles, often extending to periodic cycles. Here are key details:

• \( \sin x \) is a fundamental trigonometric function that oscillates between -1 and 1.
• \( \tan x \) is the quotient of \( \sin x \) and \( \cos x \), and it has vertical asymptotes where \( \cos x = 0 \).
• \( \sec x \), the reciprocal of \( \cos x \), also displays asymptotic behavior at the same points as \( \tan x \).

When including these functions in our infinite series, they illustrate how small changes in the angle can significantly affect the output, especially as you approach the asymptotes. This characteristic is what allows the extension of the function’s range to include all real numbers.