Question

Use the root test to determine whether \(\sum_{m=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$ a_{k}=\left(\frac{2 k^{2}-1}{k^{2}+3}\right)^{k} $$

Short answer

The series diverges.

Step-by-step solution

Understand the Root Test

The root test involves the calculation of \( \limsup_{n \to \infty} \sqrt[n]{|a_n|} \). If this limit is less than 1, the series converges absolutely; if it is greater than 1, the series diverges; and if it equals 1, the test is inconclusive.

Express the General Term

We have \( a_k = \left(\frac{2k^2 - 1}{k^2 + 3}\right)^k \). We are going to apply the root test by examining \( \sqrt[k]{\left|a_k\right|} \).

Apply the Root to the Term

Consider the expression \( \sqrt[k]{\left(\frac{2k^2 - 1}{k^2 + 3}\right)^k} \). This simplifies to \( \frac{2k^2 - 1}{k^2 + 3} \).

Simplify the Limit Expression

Now consider \( L = \lim_{k \to \infty} \left(\frac{2k^2 - 1}{k^2 + 3}\right) \). We simplify this by dividing the numerator and the denominator by \( k^2 \), giving \( L = \lim_{k \to \infty} \frac{2 - \frac{1}{k^2}}{1 + \frac{3}{k^2}} \).

Solve the Limit

As \( k \to \infty \), the terms \( \frac{1}{k^2} \to 0 \), so the expression simplifies to \( L = \frac{2}{1} = 2 \).

Conclusion from the Root Test

Since the limit \( L = 2 \) is greater than 1, by the root test, the series \( \sum_{m=1}^{\infty} a_n \) diverges.

Key concepts

Convergence and Divergence of Series

Convergence and divergence are key concepts in the analysis of infinite series. When we talk about a series converging, it means that as we sum more terms, the total amount approaches a specific finite value. On the other hand, divergence indicates that the sum does not approach a particular limit, possibly increasing indefinitely or failing to settle at a stable value.

To determine whether a series converges or diverges, mathematicians use a variety of tests. The Root Test is one method used when each term of the series is raised to the power in a form such as \(\left( x_n \right)^n\). By taking the limit of the \(n\)-th root of the absolute value of each term, we can assess whether a series converges or diverges. Specifically, if this limit is:

• Less than 1, the series converges absolutely.
• Greater than 1, the series diverges.
• Exactly 1, the test is inconclusive, so other tests might be needed.

Understanding convergence and divergence is fundamental in calculus and analysis, where it helps predict the behavior of series in mathematical models, physics, and other fields.

Limit Supremum

The concept of "limit supremum" is crucial when applying the Root Test. The limit supremum, or "lim sup", of a sequence \((a_n)\), denoted as \( \limsup_{n \to \infty} a_n \), is the greatest limit point of the sequence. This may sound abstract, but it is essentially the largest value that the terms of the sequence get arbitrarily close to as the sequence progresses.

In practical terms, when applying the Root Test, you calculate the limit of the \(k\)-th root of \( |a_k| \) as k approaches infinity. The value you arrive at will help determine the behavior of the infinite series.

• If the limit supremum is less than 1, we conclude the series converges.
• If the limit supremum is greater than 1, like in the example provided where \( L = 2 \), we ascertain divergence.

Calculating the limit supremum involves examining the highest potential accumulation point of the sequence's value as you move through infinite terms.

Infinite Series Analysis

Analyzing infinite series involves understanding their behavior and convergence properties, whether they stabilize to a finite sum or diverge. Series analysis techniques, like the Root Test, allow us to make predictions about these characteristics.

The infinite series \( \sum_{k=1}^{\infty} a_k \) can reveal a lot of information about the underlying function or pattern generating its terms. In the example, the series diverges as predicted by the Root Test. This means summing even an infinite number of terms would not yield a finite number but would rather grow indefinitely. Such discovery might indicate practical constraints or limitations in real-world situations where these mathematical models apply.

• The Root Test provides an insight into the geometric form of the terms.
• Understanding these patterns is crucial for integrating infinite series into broader mathematical and physical theories.

With a solid grasp of series analysis, mathematicians can tackle complex problems within engineering, economics, physics, and beyond, converting abstract mathematics into tangible insights.