Question

An alternating series \(\sum_{n=i}^{\infty} a_{n}\) is given. (a) Determine if the series converges or diverges. (b) Determine if \(\sum_{n=0}^{\infty}\left|a_{n}\right|\) converges or diverges. (c) If \(\sum_{n=0}^{\infty} a_{n}\) converges, determine if the convergence is conditional or absolute. $$\sum_{n=0}^{\infty}\left(-\frac{2}{3}\right)^{n}$$

Short answer

The series converges absolutely because both the original and absolute series converge.

Step-by-step solution

Identify the given series

The given series is \( \sum_{n=0}^{\infty} \left(-\frac{2}{3}\right)^{n} \). This is a geometric series with the common ratio \( r = -\frac{2}{3} \).

Use geometric series test

For a geometric series \( \sum_{n=0}^{\infty} ar^n \), the series converges if the absolute value of the common ratio \( |r| < 1 \). In this series, \( |r| = \left| -\frac{2}{3} \right| = \frac{2}{3} < 1 \), so the series converges.

Determine if the absolute series converges

The absolute series is \( \sum_{n=0}^{\infty} \left| \left(-\frac{2}{3}\right)^n \right| = \sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n \). This is also a geometric series with \( r = \frac{2}{3} < 1 \), so it converges.

Determine the type of convergence

Since both the original series \( \sum_{n=0}^{\infty} \left(-\frac{2}{3}\right)^n \) and the absolute series \( \sum_{n=0}^{\infty} \left| \left(-\frac{2}{3}\right)^n \right| \) converge, the convergence is absolute.

Key concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In other words, it looks like this: \( a, ar, ar^2, ar^3, \ldots \). The sum of the terms in a geometric sequence gives us the geometric series. An infinite geometric series is expressed as \( \sum_{n=0}^{\infty} ar^n \). One important property of geometric series is their convergence.
- If the absolute value of the common ratio \( |r| \) is less than 1, the series converges.- If \( |r| \) is equal or greater than 1, the series diverges.
In our series \( \sum_{n=0}^{\infty} \left(-\frac{2}{3}\right)^n \), the common ratio \( r \) is \( -\frac{2}{3} \). Since \( |r| = \frac{2}{3} < 1 \), it converges. Understanding this basic property of geometric series can help you greatly in assessing their convergence.

Absolute Convergence

A series \( \sum a_n \) has absolute convergence if the series of absolute values \( \sum |a_n| \) also converges. This concept is crucial because if a series is absolutely convergent, it guarantees the original series itself also converges.
Consider our given series, \( \sum_{n=0}^{\infty} \left(-\frac{2}{3}\right)^n \). To test for absolute convergence, we look at \( \sum_{n=0}^{\infty} \left| \left(-\frac{2}{3}\right)^n \right| = \sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n \).
This transforms the series into a typical geometric series with a positive ratio \( r = \frac{2}{3} \), which is less than one, thus it converges.
The convergence of \( \sum |a_n| \) assures us that our series is absolutely convergent.

Conditional Convergence

Conditional convergence occurs when a series converges, but it does not converge absolutely. That is, the series \( \sum a_n \) converges, but \( \sum |a_n| \) diverges. A common example of a conditionally convergent series is the alternating series.
For the series in our problem, \( \sum_{n=0}^{\infty} \left(-\frac{2}{3}\right)^n \), we find that not only does the series itself converge, but also the series of absolute values converges. Therefore, the series isn't just conditionally convergent; it is absolutely convergent. Recognizing the nature of each series helps in understanding how the behavior of its terms affects convergence.

Convergence Tests

Convergence tests are methods used to determine whether a series converges or diverges. Some of the most commonly used tests include the Geometric Series Test, the p-series Test, the Ratio Test, and the Alternating Series Test.
- **Geometric Series Test:** This checks if \( |r| < 1 \) for a geometric series, ensuring convergence.- **Alternating Series Test:** A series where terms alternate in sign and meet certain decreasing criteria may converge even if not absolutely.
In our case, the Geometric Series Test is applied because the structure of our series \( \sum_{n=0}^{\infty} \left(-\frac{2}{3}\right)^n \) gives away that it is geometric with \( |r| < 1 \).
Convergence tests provide a systematic approach to kitchen the behavior of infinite series and identify their convergence properties.