Problem 60 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(C) \(\frac{1}{3}\)

Step 1: Analyze the term

The given expression for each term of the series is \(t_r = \frac{r}{r^4 + r^2 + 1}\). As \(r\) becomes very large, \(r^4\) dominates over \(r^2\) and 1 in the denominator, so we can approximate \(t_r \approx \frac{r}{r^4}\).

Step 2: Simplify the term

Simplifying the expression \(\frac{r}{r^4}\), we get \(t_r \approx \frac{1}{r^3}\). This approximation works well for large values of \(r\).

Step 3: Analyze the sum

The sum \(\sum_{r=1}^{n} t_r\) can be approximated for large \(n\) by replacing each term with \(\frac{1}{r^3}\). Hence, the sum becomes approximately \(\sum_{r=1}^{n} \frac{1}{r^3}\).

Step 4: Determine the convergence

The series \(\sum_{r=1}^{\infty} \frac{1}{r^3}\) is a known convergent p-series with \(p = 3 > 1\). This means the series converges to a finite limit as \(n\) approaches infinity.

Step 5: Conclude the result

Since \(\sum_{r=1}^{n} \frac{1}{r^3}\) converges as \(n \to \infty\), the original series \(\sum_{r=1}^{n} t_r\) also converges to the same finite sum. Therefore, the limit is a finite number, fitting the option (C).

Key Concepts

Convergence of Series

Convergence in the context of series refers to whether the sum of the terms in a series approaches a specific value as the number of terms grows infinitely large. When we say a series converges, it means that as we add more and more terms, the total sum gets closer to a certain finite number. For a series to be convergent, the terms need to become very small as you go beyond a certain point.

In our exercise, the term being analyzed is the limit of the series as the number of terms approaches infinity. Initially, it's critical to simplify the individual terms properly for convergence tests. Once simplified to a known form, like a p-series, convergence can be deduced based on mathematical properties. For this exercise, the series is shown to converge because the sum \[\sum_{r=1}^{\infty} \frac{1}{r^3}\]is a p-series with a power greater than 1, ensuring convergence.

P-Series

A p-series is a specific type of infinite series where each term can be represented in the form \[\frac{1}{r^p}\]with \(r\) being a positive integer that starts at 1 and \(p\) is a real number. The behavior of the p-series depends dramatically on the value of \(p\).

• If \(p > 1\), the series converges, which means the sum of all terms approaches a finite limit.
• If \(p \leq 1\), the series diverges, meaning the sum grows indefinitely as more terms are added.

In our scenario, the term approximates to \(\frac{1}{r^3}\), clearly designating it as a p-series with \(p = 3\). Since \(p\) is greater than 1, this confirms the convergence of the series, validating the result of the exercise.

Approximation of Terms

Approximation in terms of series is a process used to simplify complex series expressions into more familiar or manageable forms. This makes it easier to assess properties like convergence or divergence. Often, for large values of the variable, certain parts of an expression dominate, allowing us to see what the most significant terms will be.

In the given exercise, the initial term is\[t_r = \frac{r}{r^4 + r^2 + 1}\]By focusing on the largest power in the denominator, namely \(r^4\), we simplify this to \[t_r \approx \frac{r}{r^4} = \frac{1}{r^3}\]This simplification is crucial as it allows us to recognize the form of a known p-series. Such approximations are a powerful tool in mathematical analysis, especially when proving convergence or identifying behaviors in limits.