Problem 64 - Essential Calculus

Question

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty}\left(\frac{4 n+3}{2 n-1}\right)^{n} $$

Step-by-Step Solution

Answer

The given series is divergent.

Step 1: Apply the Root Test

To apply the Root Test to our series, compute the limit of the nth root of the absolute value of the nth term of the series as n approaches infinity. For our series, the nth term is \((\frac{4n+3}{2n-1})^n\). Therefore, we compute the limit: \[ \lim_{{n\to\infty}}\sqrt[n]{{\left|\left(\frac{4n+3}{2n-1}\right)^n\right|}} \]

Step 2: Simplify the limit

The nth root of a term raised to the power of n simplifies the expression. So the limit simplifies to:\[ \lim_{{n\to\infty}}\left|\frac{4n+3}{2n-1}\right| \]

Step 3: Evaluate the limit

When taking limit of a rational function where the highest power of n in the numerator and denominator is the same, divide every term by \(n^{highest power}\). Here it is 1. So:\[\lim_{{n\to\infty}}\left|\frac{4+\frac{3}{n}}{2-\frac{1}{n}}\right| \]This simplifies as n goes to infinity, to:\[\left|\frac{4}{2}\right| = 2\]

Step 4: Determine convergence or divergence

As 2 is greater than 1, according to the Root Test, the series is divergent.

Key Concepts

Convergence

Convergence in the context of a series refers to a situation where the sum of the series approaches a specific value as the number of terms goes to infinity. In simpler terms, as you keep adding more and more terms of the series, the total sum gets closer to a particular number and stays around that number.
There are different tests to check for convergence, such as the Root Test, Ratio Test, and others. The Root Test, specifically, examines the nth root of the absolute value of a series' terms and helps determine whether the sum converges. If the result of the Root Test is less than 1, we say the series converges. This means our series has a finite sum and is well-behaved at infinity.

Divergence

Divergence indicates when a series does not settle to a particular value as its terms are added indefinitely. In other words, as the number of terms grows, the series keeps increasing or oscillating and does not approach any finite limit.
In the exercise at hand, we found that the result of our Root Test was 2, which is greater than 1. This result led us to declare the series as divergent. It means the sum of all terms indefinitely grows without reaching a finite sum. When a series diverges, it can often imply that it either keeps increasing indefinitely or behaves erratically without any predictable end behavior.

Series Analysis

Series analysis involves examining a series to understand its properties and behavior. By applying various tests and approaches, we determine whether a series converges, diverges, or has other notable characteristics.
The series involved in our exercise, \\(\sum_{n=1}^{\infty}\left(\frac{4 n+3}{2 n-1}\right)^{n}\), was analyzed using the Root Test. Through this analysis, we examined the long-term behavior of the series by simplifying and evaluating the limit as n approaches infinity. This process told us how the terms of the series behave as a group, rather than individually. Series analysis tools often include limits, comparisons with simpler series, or known convergence tests to discern the behavior of complex series.