91Ó°ÊÓ

In mathematics, a geometric series is a series of terms in which each term is derived by multiplying the previous term by a constant, known as the common ratio. A simple way to recognize a geometric series is by observing if there’s a pattern of multiplying the base number by a constant to generate subsequent numbers. Consider the series: \((0.7)^{1}, (0.7)^{2}, (0.7)^{3}, \ldots\), where each term is obtained by multiplying the previous term \((0.7)^{n-1}\) by 0.7. Here, 0.7 is both the first term and the constant common ratio.

The geometric series can be infinite, as shown in the exercise, meaning the terms go indefinitely. For the series to converge (sum to a finite number), the common ratio must satisfy \(-1 < r < 1\). If the ratio is outside this range, the series will diverge (not sum to a finite number). This idea is crucial for understanding why the formula \(S = \frac{a}{1 - r}\) works, where \(a\) is the initial term and \(r\) is the common ratio for \([-1 < r < 1]\) scenarios.

An infinite series is a sum of an infinite sequence of terms. Mathematicians often write it using the sigma notation \(\sum\), which symbolizes addition of a sequence. An infinite series can appear intimidating because its terms extend without end.

The primary question around infinite series is whether they converge (sum to a specific value) or diverge (do not sum to a specific value). The convergence of a series depends on the behavior of its terms as the series progresses.

• If the terms approach zero and the total sum of numbers stabilize to a finite limit, the series is convergent.
• However, if the sum keeps increasing or does not stabilize, the series is divergent.

As in the exercise, calculating the convergence of \((0.7)^n + (0.9)^n\) involves evaluating each part separately concerning their own converge conditions.

The sum of a series refers to the total when all terms are added together. When dealing with an infinite geometric series, we use the formula \(S = \frac{a}{1 - r}\) to find its sum when the series converges.

Let's take a closer look at the given exercise: \((0.7)^n\) and \((0.9)^n\) are two geometric series.

• For the first, the sum \(S_1\) is calculated as \(\frac{0.7}{1 - 0.7} = 2.3333\).
• For the second, the sum \(S_2\) is \(\frac{0.9}{1 - 0.9} = 9\).

These calculations are based on the formula, considering that both common ratios \(r_1 = 0.7\) and \(r_2 = 0.9\) are less than 1, ensuring convergence.

Finally, to determine the total sum of the series \(\sum_{n=1}^{\infty}\left[(0.7)^{n}+(0.9)^{n}\right]\), we add \(S_1\) and \(S_2\): \(2.3333 + 9 = 11.3333\). This result implies the combined effect of convergence seen in both sub-series, highlighting the crucial role of understanding the convergence criterion and proper formula application.