Problem 138 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of \(2^{1 / 4} \cdot 4^{1 / 8} \cdot 8^{1 / 16} \ldots \infty\) is: (A) 1 (B) 2 (C) \(3 / 2\) (D) 4

Step-by-Step Solution

Answer

The value is 4.

Step 1: Understand the Problem

We need to find the value of the infinite product: \(2^{1/4} \cdot 4^{1/8} \cdot 8^{1/16} \ldots\). Each term in the sequence takes the form \( (2^n)^{1/2^{n+1}} \).

Step 2: Express Each Term in Uniform Base

Notice that \(4 = 2^2\), \(8 = 2^3\), etc. So, the terms can be rewritten as \( 2^{n/2^{n+1}} \). For example, \(4^{1/8} = (2^2)^{1/8} = 2^{1/4}\).

Step 3: Combine the Exponents

By properties of exponents, combine the terms: \(2^{1/4 + 1/4 + 3/16 + \, \ldots} = 2^{\sum_{n=1}^{\infty} \frac{n}{2^{n+1}}}\).

Step 4: Evaluate the Series

Notice the series is: \[ \sum_{n=1}^{\infty} \frac{n}{2^{n+1}} = \frac{1}{2} \sum_{n=1}^{\infty} \frac{n}{2^n} \]. This is an arithmetico-geometric series whose sum is known to be 2.

Step 5: Calculate the Final Exponent

Substitute the sum of the series back: \(2^{2} = 4\).

Step 6: Conclusion

The evaluated value of the infinite product \(2^{1 / 4} \cdot 4^{1 / 8} \cdot 8^{1 / 16} \ldots \) is \(4\). Therefore, the correct option is (D) 4.

Key Concepts

Arithmetico-Geometric Series

The concept of an arithmetico-geometric series might seem complex at first glance, but it's easier if we break it down. An arithmetico-geometric series is a series that combines elements of both arithmetic and geometric progressions. In simpler terms:

• An arithmetic sequence has a constant difference between consecutive terms.
• A geometric sequence has a constant ratio between terms.

When these two types of sequences merge, you get an arithmetico-geometric series. In this problem, the infinite series we need to evaluate is \( \sum_{n=1}^{\infty} \frac{n}{2^n} \). This series is arithmetico-geometric because:

• The numerator increases arithmetically (i.e., by adding 1 each time).
• The denominator grows geometrically (i.e., multiplying by 2 each time).

This specific type of series has a known sum, which is extremely helpful: \( \sum_{n=1}^{\infty} \frac{n}{2^n} = 2 \). Understanding this fundamental concept allows us to effectively evaluate infinite products like the one given in the exercise.

Properties of Exponents

Understanding the properties of exponents is crucial for manipulating and simplifying expressions. Exponents represent how many times a number, called the base, is multiplied by itself. Here are some key properties of exponents used in this exercise:

• Power of a Power Property: \( (a^m)^n = a^{m \cdot n} \). This is often used when raising an exponent to another exponent.
• Product of Powers Property: \( a^m \cdot a^n = a^{m+n} \). This property allows us to combine bases into a single power when they have the same base.
• Base Conversion: Numbers like 4 and 8 can be expressed as powers of 2 (e.g., \( 4 = 2^2 \) and \( 8 = 2^3 \)), facilitating uniform manipulation.

In the exercise, we use these properties to rewrite expressions such as \( 4^{1/8} \) and \( 8^{1/16} \) in terms of base 2. This simplification is key to combining all terms into a single infinite product. By converting each part of the series to the same base and using the properties of exponents, the terms can be easily summed, leading to the evaluation of the series.

Series Evaluation

Evaluating series is a critical skill in mathematics that allows us to find the sum of infinitely many terms. When we evaluate a series, the goal is often to find a closed form, or a single expression, for the sum.
In this exercise, we focus on the series:\[ \sum_{n=1}^{\infty} \frac{n}{2^{n+1}} = \frac{1}{2} \sum_{n=1}^{\infty} \frac{n}{2^n} \]which has been identified as an arithmetico-geometric series. Evaluating this type of series involves recognizing the pattern and applying known formulas.
The series \( \sum_{n=1}^{\infty} \frac{n}{2^n} = 2 \) is a well-known result. This allows us to evaluate complex infinite products by reducing them to known series forms. Once the sum is known, it can be substituted back into the mathematical expression to calculate the final value of the infinite product.
Understanding the concept of series evaluation helps with myriad problems in higher mathematics, and mastering it lays a foundation for analyzing more complex mathematical concepts.