Question

Determine whether the infinite series converges or diverges. If it converges, find the sum. $$ 4+\frac{12}{5}+\frac{36}{25}+\frac{108}{125}+\ldots $$

Short answer

The series converges, and the sum is 10.

Step-by-step solution

Identify the series

Recognize that the series is geometric because each term after the first is obtained by multiplying the previous term by a common ratio, denoted as \(r\).

Determine the common ratio

Calculate the common ratio \(r\) by dividing the second term by the first term: \( r = \frac{12/5}{4} = \frac{12}{5} \times \frac{1}{4} = \frac{3}{5} \).

Check if the series converges

For a geometric series \( a + ar + ar^2 + ar^3 + \ldots \) to converge, the absolute value of the common ratio \( r \) must be less than 1 (\( |r| < 1 \)). Here, \( r = \frac{3}{5} \), which satisfies this condition.

Find the sum of the series

The sum \( S \) of an infinite geometric series can be found using the formula \( S = \frac{a}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio. Substituting \( a = 4 \) and \( r = \frac{3}{5} \), we get: \[ S = \frac{4}{1 - \frac{3}{5}} = \frac{4}{\frac{2}{5}} = 4 \times \frac{5}{2} = 10 \].

Key concepts

Infinite Series

In mathematics, an infinite series is the sum of an infinite number of terms. To understand this better, think of adding numbers together without end.
For example, in the series given in the exercise: $$4+\frac{12}{5}+\frac{36}{25}+\frac{108}{125}+\text{…}.$$
This series does not stop. We call it an ‘infinite series’ because it goes on forever.
Understanding if an infinite series converges or diverges is crucial. If a series converges, it means the sum of its terms approaches a specific value as more terms are added. If it diverges, the sum grows without bound.
Mathematicians use various tests to check whether an infinite series converges or diverges, one of which is the geometric series test.

Common Ratio

The common ratio, represented as ‘r’, is a key element in a geometric series. It is the factor by which each term is multiplied to get the next term.
In our example series, we find the common ratio by dividing the second term by the first term:
$$\frac{12/5}{4} = \frac{12}{5} \times \frac{1}{4} = \frac{3}{5}.$$
Here, the common ratio is \( \frac{3}{5} \). This means that each term in the series is multiplied by \( \frac{3}{5} \) to get the next term.
Understanding the common ratio helps us determine the behavior of the series. If the absolute value of the common ratio (|r|) is less than 1, the series might converge. If |r| is 1 or greater, the series will diverge.

Series Convergence

Convergence in the context of series means that the terms of the series are approaching a particular value as more terms are added. For a geometric series to converge, the common ratio’s absolute value must be less than 1 (<|r|<1).
In our series, the common ratio <\br> $$r = \frac{3}{5}$$
is less than 1, satisfying this condition.
This indicates that as we keep adding more terms, the total sum settles towards a finite number, showing convergence. However, simply deciding that \( |r| < 1 \) isn't enough, we need to determine what the sum converges to which leads us to the next concept.

Sum of Geometric Series

If a geometric series converges, we can find its sum using a simple formula. The sum, S, of an infinite geometric series is given by:
$$S = \frac{a}{1-r},$$
where ‘a’ is the first term and ‘r’ is the common ratio.
In our example, the first term (a) is 4 and the common ratio (r) is \( \frac{3}{5} \). Plug these values into the formula:
\[ S = \frac{4}{1 - \frac{3}{5}} = \frac{4}{\frac{2}{5}} = 4 \times \frac{5}{2} = 10. \]
Therefore, the sum of the series is 10.
This formula is powerful as it turns the daunting task of adding infinite terms into a simple calculation. Knowing the first term and the common ratio allows us to quickly find the sum if the series converges.