Problem 89 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(\lim _{n \rightarrow \infty}\left[\sum_{r=1}^{n} \frac{1}{2^{r}}\right]\), where \([\cdot]\) denotes the greatest integer, is (A) 0 (B) 1 (C) \(-1\) (D) \(\frac{1}{2}\)

Step-by-Step Solution

Answer

The value is 1.

Step 1: Identify the Series

The given series is \(\sum_{r=1}^{n} \frac{1}{2^{r}}\). This is a geometric series with the first term \(a = \frac{1}{2}\) and the common ratio \(r = \frac{1}{2}\).

Step 2: Sum of an Infinite Geometric Series

The sum \(S\) of an infinite geometric series with the first term \(a\) and common ratio \(0 < |r| < 1\) is given by \(S = \frac{a}{1 - r}\).

Step 3: Calculate the Limit of the Series

For our series, the first term \(a = \frac{1}{2}\) and the common ratio \(r = \frac{1}{2}\), so the sum is \(\frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1\).

Step 4: Evaluate the Greatest Integer Function

Since the sum of the infinite series is 1, the greatest integer \([\cdot]\) function applied to 1 is \(1\), because the greatest integer less than or equal to 1 is 1.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. It can be expressed in the form:

• First term: \(a\)
• Common ratio: \(r\)

For a finite geometric series, we can find the sum of the first \(n\) terms using the formula:\[S_n = a \frac{1-r^n}{1-r}\]where \(n\) is the number of terms, \(a\) is the first term, and \(0 < |r| < 1\). However, if the geometric series has infinitely many terms, it is called an infinite geometric series.
A geometric series converges (approaches a finite value) if the absolute value of the common ratio is less than one.

Infinite Series

An infinite series is the sum of an infinite sequence of numbers. Such series may converge, meaning they approach a finite limit, or diverge, where they do not settle at a specific value.
A well-known example of a converging infinite series is the geometric series with a common ratio satisfying \(|r| < 1\).
For instance, in the geometric series example \(\sum_{r=1}^{\infty} \frac{1}{2^r}\), the first term \(a\) is \(\frac{1}{2}\) and the common ratio \(r\) is \(\frac{1}{2}\).
The sum \(S\) of this infinite geometric series is given by:\[S = \frac{a}{1 - r} = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1\]This calculation shows how an infinite series can indeed converge to a fixed number.

Greatest Integer Function

The greatest integer function, also known as the floor function, maps a real number to the largest integer less than or equal to it.
It is denoted as \([x]\), where for any real number \(x\), \([x]\) is the greatest integer \(k\) such that \(k \leq x\).
This means if you have a value of \(1.7\), the greatest integer function will return \(1\) because it is the largest integer that is not greater than \(1.7\). Conversely, \([-2.3]\) would return \(-3\).
For the given problem, when applied to the sum of the infinite geometric series which converges to \(1\), the greatest integer function \([1]\) results in \(1\), as it is the greatest integer less than or equal to \(1\).