Problem 14 - Algebra for JEE ADVANCED

Question

Find the sum of \(n\) terms of the series whose \(n^{\text {th }}\) term is \(T(n)=\tan \frac{x}{2^{n}} \times \sec \frac{x}{2^{n-1}}\)

Step-by-Step Solution

Answer

The sum of the first $n$ terms is $\sec \frac{x}{2} - \sec \frac{x}{2^n}$.

Step 1: Analyze the General Term

The given term of the series is \( T(n) = \tan \frac{x}{2^{n}} \times \sec \frac{x}{2^{n-1}} \). Recognize that this can be transformed using trigonometric identities.

Step 2: Apply Trigonometric Identity

Using the identity \( \tan A \sec B = \sec A - \sec B \) when \( A + B = 90^\circ \), transform \( \tan \frac{x}{2^n} \sec \frac{x}{2^{n-1}} \) into terms of secant. This works due to geometric interpretation or by verifying the change under specific constraints.

Step 3: Simplify the nth Term

Re-write \( T(n) \) as \( \sec \frac{x}{2^n} - \sec \frac{x}{2^{n-1}} \). This conversion is critical because it allows the series to become telescoping.

Step 4: Formulate the Sum of the Series

The series is now \( \sum_{n=1}^{N} \left( \sec \frac{x}{2^n} - \sec \frac{x}{2^{n-1}} \right) \). This is a telescoping series where most terms cancel out.

Step 5: Evaluate the Telescoping Series

Due to the telescoping nature, all intermediate terms cancel, leaving only the first and the last terms. The sum simplifies to \( \sec \frac{x}{2^1} - \sec \frac{x}{2^N} \).

Step 6: Write the Final Expression

Deduce that the total sum of the first \( n \) terms is \( \sec \frac{x}{2} - \sec \frac{x}{2^n} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools for simplifying and transforming trigonometric expressions. They relate different trigonometric functions like sine, cosine, tangent, and secant. In this particular exercise, the identity \( \tan A \sec B = \sec A - \sec B \) is utilized. This identity is applicable when the sum of angles \( A \) and \( B \) equals \( 90^{\circ} \), illustrating how different trigonometric functions can express equivalent relationships.
Recognizing and applying trigonometric identities can often turn complex problems into simpler equations. It helps in understanding how angles and functions relate, especially in series and summation problems.
Key identities to remember include:

• Reciprocal identities: \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• Pythagorean identities: Such as \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Angle sum identities: Like \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \).

These identities serve as shortcuts that simplify the solution process by converting expressions into a more manageable form.

Secant Function

The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function. It is defined as \( \sec \theta = \frac{1}{\cos \theta} \).
In this problem, the secant function is central due to its role in transforming the series' terms. By using the identity \( \tan A \sec B = \sec A - \sec B \), \( \sec \theta \) helps reduce complex terms into a simplified telescoping series.
The secant function is critical in many trigonometric transformations and plays an important role when dealing with certain types of series. Key aspects to understand about the secant function include:

• It is undefined at angles where the cosine function is zero, such as \( 90^{\circ}, 270^{\circ} \), etc., leading to he vertical asymptotes in its graph.
• It measures the ratio of the hypotenuse to the adjacent side in a right-angled triangle interpreted on the unit circle.
• Understanding the properties of \( \sec \theta \) and its behavior at critical angles is essential for solving advanced problems in trigonometry and calculus.

Utilizing the secant function effectively can make solving complex trigonometric series manageable and clear.

Summation Techniques

Summation techniques are strategies used to find the sum of terms in a sequence or series, which can be arithmetic, geometric, or more complex patterns. For a telescoping series, these techniques are particularly useful.
In this problem, after transforming the terms using trigonometric identities, the series is recognized as telescoping. Essentially, the telescoping series is a type of series where many terms cancel out, simplifying the final sum over a large number of terms.
To identify a telescoping series:

• Rewrite the terms so that each term cancels one or more subsequent terms.
• Only a few terms remain at the end, usually the first and last terms of the rewritten sequence.

By simplifying the terms using the identity \( \sec \theta_n - \sec \theta_{n+1} \), all intermediate terms cancel. Thus, in telescoping series, the sum becomes \( \sec \frac{x}{2} - \sec \frac{x}{2^n} \) as seen in the given solution. This shows how an apparently complex summation simplifies into a difference of the first and last transformed terms.
Key summation techniques include:

• Arithmetic series: Summing a sequence of numbers where there is a constant difference between consecutive terms.
• Geometric series: Summing terms that have a constant ratio between consecutive terms.
• Telescoping series: Using term cancellation to significantly reduce the complexity of the summation.

Mastering these techniques is invaluable in efficiently solving series-based mathematical problems.