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

Question

Instructions: 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: \(\lim _{n \rightarrow \infty} \frac{1}{n}\left(1+e^{1 / n}+e^{2 / n}+\ldots+e^{(n-1) / n}\right)=e-1\) Reason: \(1+r+r^{2}+\ldots+r^{n-1}\) \(=\left\\{\begin{array}{l}\frac{1-r^{n}}{1-r} \text { if } r<1 \\\ \frac{r^{n}-1}{r-1} \text { if } r>1\end{array}\right.\)

Step-by-Step Solution

Answer

(B) Assertion is True, Reason is True; Reason is not a correct explanation for Assertion.

Step 1: Analyzing the Assertion

The assertion \[ \lim _{n \rightarrow \infty} \frac{1}{n}\left(1+e^{1 / n}+e^{2 / n}+\ldots+e^{(n-1) / n}\right)=e-1\] needs to be examined. This sequence within the limit can be interpreted as a Riemann sum representing the integral of the function \(e^x\) from \(0\) to \(1\), which is equal to \(e^1 - e^0 = e - 1\). Thus, the assertion is true.

Step 2: Analyzing the Reason

The reason given states the formula for 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 is a correct mathematical identity for a geometric series. However, to be applicable, each term of the sequence needs to form a geometric series, which is not the case here as the sequence involves \(e^{k/n}\) rather than a power of a common ratio \(r\). Thus, the reason is irrelevant to the assertion.

Step 3: Comparing Assertion and Reason

The assertion correctly points to a property of calculus linked to Riemann sums and definite integrals, whereas the reason explains a property related to geometric progressions, which is unrelated to the specific operation in this limit problem. Thus, while both the assertion and the reason are true, the reason is not a correct explanation for the assertion.

Key Concepts

Riemann Sum

A Riemann sum is a method for approximating the integral or area under a curve. Imagine you are trying to find the area under a curve on a graph. If you were using a Riemann sum, you'd break the area into small rectangles, calculate the area of each rectangle, and then sum them all together.

You can think of it like this:

• Divide the interval into small sub-intervals.
• For each sub-interval, calculate the area of a rectangle where the height is the value of the function at some point in the sub-interval.
• Add up all these areas to get an approximation of the total area under the curve.

The tighter or smaller you make these intervals, the more accurate your Riemann sum becomes. When you take the number of rectangles to infinity, what you get is the exact integral.—This connection is vital for understanding calculus, as it explains how integration is the natural extension of summing up all those little pieces.

Definite Integral

A definite integral calculates the total area under a curve between two points on the graph. It's like taking the sum of all the infinite Riemann sums within a particular interval and getting to know the exact area.

The process involves:

• Choosing a function whose area you want to calculate.
• Determining the limits of your function, i.e., the interval between which you want the area.
• Using calculus to determine this area, often represented by the integral symbol \(\int\).

The result of a definite integral is a number that represents the precise area. It’s this concept that lays the groundwork for many applications in physics, engineering, economics, and beyond. It allows you to not only estimate the area under curves but also find accumulated quantities, like distance from velocity or mass from density.

Geometric Series

A geometric series is a sum of terms where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, in the series \(1 + r + r^2 + r^3 + \ldots\), \(r\) is the common ratio.

Key properties of geometric series include:

• If the absolute value of the common ratio \(|r| < 1\), the series converges to a particular sum. The formula for this sum is \(\frac{1 - r^n}{1 - r}\) for a finite series and \(\frac{1}{1-r}\) for an infinite series.
• If \(|r| > 1\), the series diverges, which means it grows infinitely large and does not sum to a particular value.
• The formula mentioned only applies if each term in the sequence is a power of the common ratio \(r\).

Geometric series are seen in various mathematical and real-world contexts, such as calculating interest in finance or analyzing patterns in nature. They're distinct from what's happening in the Riemann sum example we discussed earlier, as those terms do not follow a geometric sequence.