Question

Let \(\sum_{n=1}^{\infty} a_{n}\) be a series with nonzero terms. Let $$ \rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| $$ i. If \(0 \leq \rho<1\), then \(\sum_{n=1}^{\infty} a_{n}\) converges absolutely. ii. If \(\rho>1\) or \(\rho=\infty\), then \(\sum_{n=1}^{\infty} a_{n}\) diverges. iii. If \(\rho=1\), the test does not provide any information.

Short answer

The series converges for \(\rho < 1\), diverges for \(\rho > 1\) or \(\rho = \infty\), and is inconclusive for \(\rho = 1\).

Step-by-step solution

Understanding the Ratio Test

The ratio test is a method used to determine the convergence or divergence of a series \(\sum a_n\). It involves taking the limit \(\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \) and checking its value to decide whether the series converges or diverges.

Applying the Test for \(0 \leq \rho < 1\)

When \(\rho < 1\), the ratio of consecutive terms \(\frac{a_{n+1}}{a_n}\) tends to decrease and the terms get smaller as \(n\) increases. This indicates that the series \(\sum a_n\) will converge absolutely, meaning it converges regardless of whether each term is positive or negative.

Analyzing the Case \(\rho > 1\) or \(\rho = \infty\)

If \(\rho > 1\), the terms \(|a_{n+1}|\) are growing larger compared to \(|a_n|\) as \(n\) increases, indicating that the series \(\sum a_n\) diverges. This is because the terms do not approach zero, and the series cannot sum to a finite value. If \(\rho = \infty\), the series diverges even more strongly for the same reason.

Understanding the Case \(\rho = 1\)

When \(\rho = 1\), the ratio test does not provide any conclusion about the series' convergence or divergence. Other convergence tests would need to be used to determine the behavior of the series in this case.

Key concepts

Convergent Series

A convergent series is one where the sum of its terms approaches a finite limit as you continue to add more terms. This is important in the study of infinite series because it tells us whether the series will settle at a particular value or continue to grow without bound.

In mathematical terms, a series \( \sum_{n=1}^{\infty} a_n \) converges if the sequence of partial sums \( S_N = a_1 + a_2 + \ldots + a_N \) approaches a specific number as \( N \to \infty \).

For example, the geometric series \( \sum_{n=0}^{\infty} r^n \) converges if \( |r| < 1 \) and its sum is \( \frac{1}{1-r} \). In this manner, the Ratio Test provides a convenient and powerful tool to check if a series is convergent by examining how the terms of the series behave as \( n \) tends to infinity.

Divergent Series

A series is said to be divergent if it does not converge, meaning its partial sums do not approach a finite limit but rather increase indefinitely or oscillate without settling at a particular value.

Divergence occurs when the terms of a series do not get smaller fast enough. With the Ratio Test, if the limit \( \rho > 1 \), it implies that the terms are increasing, leading the series to diverge. The terms of the series do not decrease sufficiently to yield a convergent sum.

Another example of a divergent series is the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \). Here, the terms \( \frac{1}{n} \) do not decrease quickly enough to sum to a finite number, demonstrating that not all series whose terms approach zero are convergent.

Absolute Convergence

When a series converges even if all its terms are replaced by their absolute values, the series is said to have absolute convergence. This is a stronger form of convergence.

For a series, absolute convergence implies that rearranging the terms will still result in the series converging to the same sum. The Ratio Test indicates absolute convergence when \( 0 \leq \rho < 1 \).

Absolute convergence has important implications, as a series that is absolutely convergent is definitely convergent, though the reverse is not always true. For instance, consider the alternating harmonic series \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n} \), which is conditionally convergent but not absolutely convergent.

Series Convergence Criteria

There are various criteria and tests that can be applied to establish whether a series converges. The Ratio Test, as mentioned, is one such criterion and is particularly useful for series involving exponentials and factorials.

To apply the Ratio Test, compute \( \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

• If \( 0 \leq \rho < 1 \), then the series converges absolutely.
• If \( \rho > 1 \) or \( \rho = \infty \), the series diverges.
• If \( \rho = 1 \), the test is inconclusive and other tests must be used.

Other tests include the Comparison Test, Integral Test, and Alternating Series Test, each useful in different contexts. The choice of test often depends on the nature of the series and its terms, making a familiarity with multiple tests beneficial for analyzing series convergence.