Problem 15 - Thomas Calculus

Question

In Exercises \(15-18,\) determine if the geometric series converges or diverges. If a series converges, find its sum. $$ 1+\left(\frac{2}{5}\right)+\left(\frac{2}{5}\right)^{2}+\left(\frac{2}{5}\right)^{3}+\left(\frac{2}{5}\right)^{4}+\cdots $$

Step-by-Step Solution

Answer

The series converges with a sum of \(\frac{5}{3}\).

Step 1: Identify the Geometric Series Parameters

First, recognize that the series is in the form of a geometric series. The general form of a geometric series is \(a + ar + ar^2 + ar^3 + \cdots \). In this particular series, \(a = 1\) and \(r = \frac{2}{5}\).

Step 2: Determine Convergence or Divergence

A geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. Here, \(r = \frac{2}{5}\) and \(|r| = \frac{2}{5} < 1\). Therefore, this series converges.

Step 3: Calculate the Sum of the Convergent Series

For a convergent geometric series, the sum \(S\) can be calculated using the formula: \[ S = \frac{a}{1-r} \] Substitute \(a = 1\) and \(r = \frac{2}{5}\) into the formula:\[S = \frac{1}{1 - \frac{2}{5}} = \frac{1}{\frac{3}{5}} = \frac{5}{3}\]

Step 4: Provide Final Answer

Having verified that the series converges and calculated its sum, the geometric series converges with a sum of \(\frac{5}{3}\).

Key Concepts

Series Convergence

Series convergence is an important concept in mathematics when examining infinite series. It helps us determine if adding an infinite number of terms will yield a finite sum. In general, a series converges when the sum of its terms approaches a specific value.

• A series is divergent if the sum does not approach any limit and instead heads toward infinity.
• To check for convergence, we often look at the behavior of the series’ terms as the series progresses.

In geometric series, convergence is particularly easy to determine using the common ratio, denoted as \(r\). If the absolute value of \(r\) is less than 1, the series will converge. This is because each term becomes progressively smaller, auguring a point where the total sum stops growing infinitely.
For the given series: \(1 + \left(\frac{2}{5}\right) + \left(\frac{2}{5}\right)^2 + \left(\frac{2}{5}\right)^3 + \cdots\), the common ratio is \(r = \frac{2}{5}\). Since \(|\frac{2}{5}| < 1\), we know this series converges.

Geometric Sequence

Understanding a geometric sequence is an essential step when working with a geometric series. A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio.
The series in this exercise follows the form of a geometric sequence:

• The first term (\(a\)) is \(1\), and represents the starting point of the series.
• The second term is \(ar\), where \(r = \frac{2}{5}\).
• This pattern continues with terms \(ar^2, ar^3, ar^4,\ldots\).

Example terms from the given sequence include \(1, \frac{2}{5}, \left(\frac{2}{5}\right)^2, \left(\frac{2}{5}\right)^3,\) and so forth. Understanding this repetitive multiplication helps in identifying the series’ nature and solving related problems.

Sum of a Series

The sum of a series is perhaps the most intriguing part for those studying sequences and series. For a converging geometric series, there is a specific formula used to find its sum:

• The formula \( S = \frac{a}{1-r} \) allows for a quick calculation of the infinite series' sum once convergence is confirmed.
• Substitute the values of the first term \(a\) and the common ratio \(r\) into this formula to get the sum.

In the provided example:
- First term \(a = 1\)
- Common ratio \(r = \frac{2}{5}\)
Substituting these values results in:\[ S = \frac{1}{1-\frac{2}{5}} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \]This calculation shows that the sum of the infinite series is \(\frac{5}{3}\). Recognizing the summed value allows us to understand complex problems and apply these concepts to real-world mathematics.