Analyzing the Given Series for Convergence

The question asks about the nature of the given series: \(1 + \frac{2}{3} + {\left( {\frac{2}{3}} \right)^2} + ... + {\left( {\frac{2}{3}} \right)^{n - 1}}\). This series is a sum of terms where each term is obtained by multiplying the previous term by a fixed number. This type of series is known as a geometric series.

Understanding Geometric Series and Convergence

A geometric series is a series with a constant ratio between successive terms. The general form of a geometric series is:

\(a + ar + ar^2 + ar^3 + ...\)

where \(a\) is the first term and \(r\) is the common ratio.

For an infinite geometric series, the behavior regarding convergence depends entirely on the value of the common ratio \(r\).

• If \(|r| < 1\), the geometric series is convergent. This means the sum of the infinite series approaches a finite value, specifically \(\frac{a}{1-r}\).
• If \(|r| \ge 1\), the geometric series is divergent. The sum does not approach a finite value.
• An oscillatory series is one that neither converges nor diverges but fluctuates between values. This can happen with some types of series, but for a geometric series, it's determined by the \(|r|\) value.

Applying the Convergence Test to the Series

Let's examine the given series: \(1 + \frac{2}{3} + {\left( {\frac{2}{3}} \right)^2} + ... + {\left( {\frac{2}{3}} \right)^{n - 1}}\). Although written to \(n-1\) terms, the options (convergent, divergent) strongly suggest considering the infinite form of this geometric series: \(1 + \frac{2}{3} + {\left( {\frac{2}{3}} \right)^2} + ... \)

Comparing this to the general form \(a + ar + ar^2 + ...\):

• The first term \(a = 1\).
• The common ratio \(r = \frac{2}{3}\) (since \(\frac{2/3}{1} = \frac{2}{3}\), and \(\frac{(2/3)^2}{2/3} = \frac{2}{3}\)).

Now, we check the condition for convergence based on the common ratio \(r\).

We need to evaluate \(|r|\):

\(|r| = \left|\frac{2}{3}\right| = \frac{2}{3}\)

Since \(\frac{2}{3}\) is less than 1 (\(0.666... < 1\)), the condition \(|r| < 1\) is satisfied.

Therefore, the infinite geometric series \(1 + \frac{2}{3} + {\left( {\frac{2}{3}} \right)^2} + ... \) is a convergent series.

The convergence of this series means that if we add up infinitely many terms following this pattern, the sum approaches a specific finite number. The sum of this convergent series can be calculated using the formula \(S = \frac{a}{1-r}\):

\(S = \frac{1}{1 - \frac{2}{3}} = \frac{1}{\frac{3-2}{3}} = \frac{1}{\frac{1}{3}} = 3\)

So, the sum of this convergent series is 3.

Conclusion on the Series Type

Based on the analysis of its common ratio, the given series is a geometric series with \(|r| < 1\). This definitively classifies it as a convergent series.

The options are:

1. convergent
2. divergent
3. oscillatory
4. None of these

Since our analysis shows the series is convergent, option 1 is the correct description.

A divergent series would have \(|r| \ge 1\). An oscillatory series behaves differently, not typically for a geometric series with a real common ratio.