Question

Find the sum of the infinite geometric series, if it exists. $$ \sum_{i=1}^{\infty} \frac{2}{5}\left(\frac{5}{3}\right)^{i-1} $$

Short answer

The sum of the given infinite geometric series does not exist because the series does not converge.

Step-by-step solution

Determine if the series is geometric

The series \(\sum_{i=1}^{\infty} \frac{2}{5}\left(\frac{5}{3}\right)^{i-1}\) looks like a geometric series, with the initial term a = \(\frac{2}{5}\) and the common ratio r = \(\frac{5}{3}\)

Check the convergence

An infinite geometric series converges if the absolute value of the common ratio r < 1. Here, \(|\frac{5}{3}|\) is greater than 1, which means the series diverges.

State the conclusion

The sum of the given infinite geometric series does not exist because the series diverges.

Key concepts

Convergence of Series

Understanding the convergence of a series is essential when we're dealing with infinite sequences. A series converges when its terms approach a specific value as the sequence progresses to infinity. In the context of the given exercise, we're tasked with checking if an infinite geometric series converges. To do this, we look at the absolute value of the common ratio of the series. If this value is less than 1, the series converges; otherwise, it diverges. In the series in question, the common ratio's absolute value is greater than 1, leading to the conclusion that the series does not converge. The importance of this concept lies in distinguishing between series that yield a finite sum and those that grow without bounds.

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This exercise involves a geometric series, which is evident from the consistent multiplication of each term by the same factor. The defining feature of a geometric series is this multiplicative pattern, which can be harnessed to find the sum of a finite series or, under the right conditions, an infinite one. It's fundamental for students to recognize this pattern to apply the correct formulas and techniques for summation.

Divergence of Series

In contrast to convergence, a series diverges when its terms do not approach a specific value and instead increase or decrease without limit. The series presented in the exercise diverges because the absolute value of the common ratio exceeds 1. Identifying divergence quickly prevents the unnecessary calculation of a sum that doesn't exist. Knowing the behavior of a series is pivotal in many areas of mathematics and its applications, since it helps predict and understand long-term behavior of sequences over time.

Common Ratio

The common ratio is the factor by which consecutive terms of a geometric series are multiplied to obtain the next term. This exercise spotlights the common ratio, \(\frac{5}{3}\), as the determining factor for the series' convergence or divergence. The common ratio is paramount because it dictates how the terms of the series evolve. A common ratio greater than 1 in absolute value means each term will be larger than the previous one, leading to divergence, while a value less than 1 indicates convergence with decreasing term sizes.

Algebraic Sequences and Series

Algebraic sequences and series refer to ordered lists of numbers and the sum of elements within these lists, respectively, following a specific formulaic pattern. In this problem, an infinite geometric series is a type of algebraic series composed of a sequence with a multiplication rule. Grasping the overall concept of algebraic sequences and series is fundamental for students as they apply to various mathematical and real-world scenarios, from calculating compound interest to analyzing sound waves. The geometric series is just one example within the broad category of algebraic series, highlighting the diverse nature of sequences and their associated sums.