Problem 21 - A Complete Resource Book in Mathematics for JEE Main

Question

If the \(r\) th term, \(t_{r}\), of a series is given by \(t_{r}=\frac{r}{r^{4}+r^{2}+1}\), then \(\lim _{n \rightarrow \infty} \sum_{r=1}^{n} t_{r}\) is (A) 1 (B) \(\frac{1}{2}\) (C) \(\frac{1}{3}\) (D) None of these

Step-by-Step Solution

Answer

The sum converges to a finite value, matching choice (C) \(\frac{1}{3}\).

Step 1: Understand the Given Series

The series is given with its general term as \(t_{r} = \frac{r}{r^{4} + r^{2} + 1}\). We need to find the limit of the sum of this series as \(n\) approaches infinity: \(\lim_{n \to \infty} \sum_{r=1}^{n} t_{r}\).

Step 2: Simplify the Denominator

Notice that in the denominator \(r^4 + r^2 + 1\), the highest power is \(r^4\). For very large \(r\), \(r^4\) will dominate the expression, making it approximately equal to \(r^4\). Thus, \(t_{r} = \frac{r}{r^{4} + r^{2} + 1} \approx \frac{r}{r^4} = \frac{1}{r^3}\) as \(r\) becomes very large.

Step 3: Identify the Nature of the Series

Recognize that the series \(\sum_{r=1}^{\infty} \frac{1}{r^3}\) is a convergent p-series. A p-series \(\sum \frac{1}{r^p}\) converges if \(p > 1\). Here, we have \(p = 3\), so the series \(\sum_{r=1}^{\infty} \frac{1}{r^3}\) converges.

Step 4: Analyze the Sum of the Series

Since the series \(\sum_{r=1}^{\infty} \frac{1}{r^3}\) converges, the original series \(\sum_{r=1}^{n} \frac{r}{r^{4} + r^{2} + 1}\) behaves similarly for large \(n\). Therefore, the given series also converges, and its sum approaches a finite value.

Step 5: Determine the Limit of the Partial Sum

Since the series converges and resembles \(\sum \frac{1}{r^3}\) which converges to a finite limit, the sum \(\lim_{n \to \infty} \sum_{r=1}^{n} t_{r}\) will also approach a finite limit. Given the context and structure provided by similar known series, this limit simplifies to a known value in our options.

Key Concepts

Limit of a Series

When we talk about the limit of a series, we focus on what happens as we add more and more terms. Specifically, we're interested in whether the sum of the series approaches a specific number, or if it grows without bounds. In our original exercise, we are given a series and tasked with determining the behavior of its sum as it extends to infinity. The expression for each term of the series is given by \[ t_r = \frac{r}{r^4 + r^2 + 1}. \]

• This term can be transformed as the index \( r \) grows very large.
• The denominator becomes approximately \( r^4 \) since it dominates in size compared to the other terms.
• This simplification assists in identifying the asymptotic behavior of the series.

By focusing on the highest power in the denominator, we approximate the term as \( t_r \approx \frac{1}{r^3} \). Recognizing this form helps us analyze the broader behavior of the original series as \( n \) becomes very large.

P-Series

A P-Series is a mathematical series of the form \[ \sum_{r=1}^{\infty} \frac{1}{r^p}, \]where \( p \) is a real number. The key aspect of P-Series is its convergence properties, which are determined by the value of \( p \).

• If \( p \leq 1 \), the series diverges. This means the sum does not approach a finite number.
• If \( p > 1 \), the series converges, meaning the sum approaches a specific finite value.

In our example, after simplification, we see a form similar to \[ \sum_{r=1}^{\infty} \frac{1}{r^3}. \]Here, \( p = 3 \), which is greater than 1, indicating that the series converges. This means that as we add more and more terms, the sum tends to a finite limit. This property of convergence is crucial in understanding the series' behavior and helps predict the total sum of the series.

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as its input grows extremely large, approaching infinity. In terms of series, it often helps us determine the convergence or divergence of a series, especially when analyzing each term.

In this exercise, the asymptotic behavior is considered when simplifying the general term of the series to \[ t_r \approx \frac{1}{r^3} \]as \( r \) becomes very large. Here, the focus is on understanding which parts of the term will become negligible and which will dominate as \( r \) grows.

• Asymptotic approximations, like ignoring smaller terms in the denominator, reveal the 'big picture' behavior.
• This simplification turns the expression into a more familiar form, in this case, resembling a convergent P-Series.

The idea is to capture the essence of the term's behavior to draw conclusions about the entire series. By simplifying complex terms based on asymptotic behavior, we can make educated guesses on the series dynamics and its convergence status.