91Ó°ÊÓ

Understanding when a geometric series converges is crucial for analyzing infinite series. A geometric series can be written as \( a + ar + ar^2 + ar^3 + \ldots \), where \( a \) is the first term and \( r \) is the common ratio. The convergence of a geometric series depends on the absolute value of the common ratio. To determine if a series converges, we check the condition \( |r| < 1 \). If this condition is met, the series is convergent, meaning the sum approaches a specific finite value as the number of terms goes to infinity.
Let's use some examples to illustrate this principle. In the first series from our exercise, \( 2+\frac{2}{3}+\frac{2}{9}+\ldots \), the common ratio is \( \frac{1}{3} \), which satisfies \( |\frac{1}{3}| < 1 \). Therefore, this first series is convergent. Another example is the second series, \( 4-2+1-\frac{1}{2}+\ldots \), with a common ratio \( -\frac{1}{2} \). Despite being negative, because \( |-\frac{1}{2}| = \frac{1}{2} < 1 \), this series is also convergent.
To sum up, for any series to converge, always examine the absolute value of the common ratio. If less than one, convergence is guaranteed.

In geometric series, the common ratio \( r \) is the factor by which each term in the series is multiplied to get the subsequent term. This ratio is an essential element that dictates the behavior of the series.

To identify the common ratio, examine any term of the series divided by the previous term. For instance, in the series \( 2+\frac{2}{3}+\frac{2}{9}+\ldots \), the second term divided by the first term gives the common ratio \( \frac{1}{3} \). Similarly, the sequence \( 4-2+1-\frac{1}{2}+\ldots \) has a common ratio of \( -\frac{1}{2} \) seen by dividing the second term \(-2\) by the first term \(4\).

• If \( |r| < 1 \), the series converges.
• If \( |r| = 1 \), the series does not converge since it does not settle at a sum.
• If \( |r| > 1 \), the series diverges.

Identifying and analyzing the common ratio allows us to easily determine whether a series will converge or diverge.

Understanding when a series diverges is just as important as understanding convergence. Divergence occurs when the sum of the series does not approach any finite number as more terms are added.

In geometric series, divergence is determined by the common ratio \( r \). Specifically, a series diverges if the absolute value of the common ratio is greater than one, \( |r| > 1 \). An example from our exercise is the series \( 10+11+\frac{121}{10}+\frac{1331}{100}+\ldots \), where \( r = \frac{11}{10} \). The absolute value of this ratio is greater than one, causing the series to diverge. Furthermore, the series \( 1-\frac{5}{4}+\frac{25}{16}-\frac{125}{64}+\ldots \), with \( r = -\frac{5}{4} \), also diverges since \( |\frac{5}{4}| > 1 \).

Diverging series lack a finite sum and essentially run off to infinity (if they're positive) or accumulate to an undefined sum. This characteristic makes it crucial to analyze the common ratio, as it determines the very nature of the series' behavior.