Question

Suppose that \(\left|\frac{a_{n+2}}{a_{n}}\right| \leq r<1\) for all \(n\). Can you conclude that \(\sum_{n=1}^{\infty} a_{n}\) converges?

Short answer

Yes, \( \sum_{n=1}^{\infty} a_n \) converges due to the generalized Ratio Test.

Step-by-step solution

Understanding the Given Condition

We are given the condition \( \left| \frac{a_{n+2}}{a_{n}} \right| \leq r < 1 \) for all \(n\). This means that every term after multiplying by two consecutive indices has a bound that's a fraction (\(r\)) of the previous terms, indicating the terms of the sequence become smaller as \(n\) increases.

Relate with a Known Convergence Test

The question implies a decrease similar to what is required by the Ratio Test for series convergence. The Ratio Test states that if \( \left| \frac{a_{n+1}}{a_{n}} \right| \leq r < 1 \) eventually (for all sufficiently large \(n\)), then the series \( \sum a_n \) converges absolutely.

Apply the Generalization for Larger Gaps

In this problem, we have a step of two indices (i.e., \(a_{n+2}\) instead of \(a_{n+1}\)). However, similar logic can be applied. Specifically, the condition given implies that if \( \frac{a_{n+2}}{a_{n}} \) is small, eventually the sequence values must decrease as well, still meeting the ultimate requirement of the Ratio Test. This shall allow concluding convergence under a broader assumption.

Conclude based on Generalization

Therefore, because this condition can be translated into terms supported by the broader application of series convergence tests, we can conclude that the series \( \sum_{n=1}^{\infty} a_n \) does converge. The condition is a generalized form of the Ratio Test for convergence.

Key concepts

Ratio Test

The Ratio Test is a powerful tool used to determine whether an infinite series converges. This test involves the terms of the series, denoted by \(a_n\). When applying the Ratio Test, we generally calculate

• \(\lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right|\)

If this limit is less than 1, the series \(\sum a_n\) converges absolutely. If the limit is greater than 1 or if it is infinite, the series diverges. Should the limit be exactly 1, the test is inconclusive, and other methods may be needed for a conclusion.

The Ratio Test simplifies analysis by focusing on the behavior of the series at its tail end. In many cases, especially when dealing with sequences involving factorials or exponential terms, this test provides clear insights into convergence.

Series Convergence

Series convergence refers to the behavior of a series as the number of terms grows indefinitely. When a series converges, it means that the sum of its infinite terms approaches a finite limit.

• A series can converge absolutely if its terms' absolute values sum to a finite number.
• Conditional convergence implies the series converges, but if we consider the absolute values of the terms, it would diverge.

Recognizing convergence is crucial, especially when working with mathematical models or calculations where precision and predictability are needed.

Different tests beyond the Ratio Test are used to determine convergence, including the Root Test, Integral Test, and Comparison Test. Each of these offers specific criteria and scenarios to aid in establishing the nature of convergence for various types of series.

Infinite Series

An infinite series is a sum of infinitely many terms, expressed typically in the form \(\sum_{n=1}^{\infty} a_n\). The concept of infinite series is fundamental in mathematics, providing the basis for complex calculations and analyses across different fields.

• The behavior of an infinite series depends heavily on the sequence of its terms \(a_n\).
• Some series, like geometric or arithmetic, have well-known convergence properties.
• Other series might necessitate intricate calculations or tests to deduce convergence or divergence.

Infinite series develop intuitions about limits and distributions, and their applications extend to calculus, physics, and several engineering domains.

Understanding infinite series involves mastering the ability to manipulate series algebraically and apply analytical tests that reveal their long-term behavior.