Problem 121 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In the following questions an Assertion \((A)\) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \(\lim _{n \rightarrow \infty} \frac{1}{n}\left(1+e^{1 n}+e^{2 / n}+\ldots+e^{(n-1) \hat{n}}\right)=e-1\) Reason: \(1+r+r^{2}+\ldots+r^{n-1}\) \(= \begin{cases}\frac{1-r^{n}}{1-r} & \text { if } r<1 \\ \frac{r^{n}-1}{r-1} & \text { if } \quad r>1\end{cases}\)

Step-by-Step Solution

Answer

Option (D): Assertion(A) is False, Reason(R) is True.

Step 1: Analyzing the Assertion

We need to determine whether the assertion \( \lim _{n \rightarrow \infty} \frac{1}{n}\left(1+e^{1 n}+e^{2 / n}+\ldots+e^{(n-1) \hat{n}}\right)=e-1 \) is true. The expression inside the limit is a sum that is divided by \( n \), where each term seems to involve exponential functions. However, upon further inspection, the expression and sequence do not seem to suggest that it simplifies to \( e-1 \), so the assertion is actually false.

Step 2: Evaluating the Reason

The given reason is about the sum of a geometric series: \( 1+r+r^{2}+\ldots+r^{n-1} = \begin{cases}\frac{1-r^{n}}{1-r} & \text {if } r<1 \ \frac{r^{n}-1}{r-1} & \text{if } r>1\end{cases} \). This statement is a standard formula for the sum of a geometric series, depending on whether \( r < 1 \) or \( r > 1 \). Therefore, the reason provided is true as it directly follows the formula for a geometric series.

Step 3: Comparing Assertion and Reason

From Step 1, we found that the assertion is false, and from Step 2, we found that the reason is true. Because the assertion is false, the logical relationship between the assertion and reason does not hold. We conclude that the correct response is option (D): Assertion(A) is False, Reason(R) is True.

Key Concepts

Limit of a function

The concept of the limit of a function is central in calculus, describing the value that a function approaches as the input approaches some value. Limits are foundational for understanding continuous functions and changes in rates. In the context of the given exercise, we were tasked with evaluating the behavior of a complex expression as n approaches infinity. This is a powerful way to analyze infinite processes and approximate the value a function is tending towards without necessarily reaching it. When working with limits:

• Identify the function and the point at which you want to find its limit.
• Consider the values the function takes as it gets closer to the point from the left and right.
• Use algebraic manipulations where necessary to simplify the function.
• Apply limit laws and techniques such as L'Hôpital's rule, if applicable.

In the exercise, we deduced that the expression does not simplify to the assumed limit, highlighting the importance of careful analysis and verification in mathematical reasoning.

Geometric series

A geometric series 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. It's an essential concept in both mathematics and finance, often used to model exponential growth or decay. The formula for the sum of a geometric series is given by different expressions depending on the magnitude of the common ratio, r:

• If \( |r|<1 \), the sum \( S_n \) is calculated as \( \frac{1-r^n}{1-r} \).
• If \( r>1 \), then \( S_n = \frac{r^n-1}{r-1} \).

This formula offers a straightforward way to determine the sum up to any finite number of terms, and is particularly powerful in determining the sum to infinity when \( |r|<1 \).In our exercise, the reason provided explained the formula for a geometric series. This formula is crucial for understanding patterns and series in calculus and solving problems involving infinite series.

Exponential functions

Exponential functions are a type of function that involves a constant base raised to a variable exponent. Commonly denoted as \( e^x \), where \( e \) is approximately equal to 2.71828, they form the basis of continuous compound growth processes. These functions are characterized by their unique property of having a rate of change proportional to the function's current value. This makes them indispensable in modeling natural growth processes, finance, and complex systems.Key points about exponential functions:

• The function \( f(x) = e^x \) is increasing and continuous for all real numbers.
• The derivative of \( e^x \) is \( e^x \) itself, reflecting the constant proportional growth rate.
• Exponential functions have a horizontal asymptote at y = 0, meaning they never actually reach zero but get infinitely close.
• They are widely used in compound interest calculations, population growth models, and natural phenomena modeling.

In our exercise, understanding exponential functions helped in dissecting the terms within the expression involving exponential growth components, which played an important role in solving complex calculus problems.