Question

In Exercises \(7-14,\) verify that the infinite series diverges. $$ \sum_{n=0}^{\infty} 1000(1.055)^{n} $$

Short answer

The given infinite series diverges because the common ratio is not between -1 and 1.

Step-by-step solution

Understand the given series

Recognize the series as a geometric series with first term (a) 1000 and the common ratio (r) 1.055. The general form of a geometric series is \( \sum_{n=0}^{\infty} ar^n\). Verify that the series fits this form.

Apply the rule for geometric series

Apply the rule which states the series converges if and only if the common ratio \(r\) is between -1 and 1. In this case, the common ratio \(r = 1.055\) which is not inside the range (-1, 1)

Draw the conclusion

Since the common ratio \(r = 1.055\) which does not lie between -1 and 1, it can be concluded that the series diverges.

Key concepts

Infinite Series

An infinite series is essentially the sum of the terms in an infinite sequence. A classic example that creates much of the groundwork for analyzing such series is the geometric series. The infinite series we often encounter in mathematics consist of simple patterns where each term can be generated from the previous one through multiplication or addition. But an important question arises: does the sum of an endlessly growing sequence of numbers actually reach a finite value, or does it keep growing indefinitely?

For instance, let's consider summing a fraction, such as \(\frac{1}{2}\), repeatedly. The first few sums might be \(\frac{1}{2}\), \(\frac{3}{4}\), \(\frac{7}{8}\), and so forth. As we add more and more terms, we get closer to the number 1, but we never actually reach it, though we can get as close as we desire. This is an example of an infinite series that converges to a finite value—in this case, 1. Conversely, the series we're analyzing involves continuously adding larger numbers, which indicates that the sum may not settle on a finite number, and instead could extend without end—an instance of divergence.

Common Ratio

In a geometric series, the common ratio (denoted as \(r\)) is the factor by which we multiply each term to get the next term in the series. It is a critical component in determining the behavior of the overall series. For the series given in the exercise, which takes the form \( \sum_{n=0}^{\infty} 1000(1.055)^{n} \), 1.055 is the common ratio. This means every subsequent term is 1.055 times the size of the preceding term.

If the common ratio is greater than 1, as in this case, each added term is larger than the last, hence, the sum of the series grows without bound - it diverges. Alternatively, if the common ratio were less than 1, then the terms would get smaller and smaller, potentially allowing the sum of the series to converge to a finite number. A common ratio of 1 would mean all terms are equal, and consequently, adding infinitely many identical terms would also lead to divergence. In summary, the value of the common ratio holds the key to understanding whether a geometric series converges or diverges.

Series Convergence Criteria

For a geometric series to converge to a finite value, it must meet certain criteria. Primarily, the series convergence criteria hinge on the absolute value of the common ratio \(r\). If \(|r| < 1\), the series has a chance to converge; if not, the series diverges. Intuitively, when \(|r| < 1\), each term in the series becomes smaller as we progress further along the series, which means that eventually, the additional terms add virtually nothing to the total sum, allowing it to stabilize at a finite value.

Testing Convergence with the Ratio Test

One method to test for convergence is the ratio test, which involves taking the limit of the absolute value of the ratio of successive terms. If this limit is less than one, the series converges. On the contrary, if the limit is greater than one, as is the case with our exercise series where the ratio is 1.055, the series definitely diverges. The ratio test provides a powerful tool for determining the behavior of series, especially when dealing with complex algebraic expressions or functions.