91Ó°ÊÓ

The concept of convergence is crucial when dealing with infinite geometric series. To determine if a series converges, we examine its common ratio. An infinite geometric series is said to converge if the absolute value of its common ratio, denoted by \(|r|\), is less than 1. This characteristic ensures that as more terms are added to the sequence, they become progressively smaller, approaching zero.

If the series converges, this leads to a finite sum, which means we can actually calculate the total sum of the series. Conversely, if \(|r|\) is greater than or equal to 1, the series diverges, and the sum cannot be determined. In our exercise, since the common ratio is \(-0.1\) and its absolute value is 0.1, which is less than 1, the series converges.

The common ratio is the factor that we repeatedly multiply by to get from one term in a geometric series to the next. It's a defining feature of a geometric series. To find the common ratio \(r\), divide any term by the preceding term.

In the exercise series \(1, -0.1, 0.01, -0.001, \ldots\), the second term divided by the first term \(-0.1 / 1\) gives us \(-0.1\) as the common ratio. It's essential to pay attention to the sign of the ratio since it affects the nature of the series—alternating positive and negative terms if the ratio is negative.

When a series converges, we can calculate its sum using a specific formula. For an infinite geometric series, the sum \(S\) is given by the expression:\[ S = \frac{a}{1 - r} \]where \(a\) is the first term and \(r\) is the common ratio.

In our example, the first term \(a\) is 1, and the common ratio \(r\) is \(-0.1\). Plugging these values into the formula, the sum is:\[ S = \frac{1}{1 - (-0.1)} = \frac{1}{1 + 0.1} = \frac{1}{1.1} \]

Calculating \(\frac{1}{1.1}\) gives approximately 0.9091, the sum of this converging series.

A geometric sequence is a set of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio \(r\). This kind of sequence is marked by its uniform structure and is straightforward to manipulate using algebraic techniques.

Each term grows or shrinks by a constant factor in a predictable manner, which is why geometric sequences are so useful in mathematical problems and real-world applications. In the provided exercise, the sequence starts at 1, and each subsequent term is obtained by multiplying the previous term by \(-0.1\), forming an alternating pattern of signs and decreasing magnitudes.