Problem 57 - Algebra and Trigonometry

Question

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 2-\frac{1}{2}+\frac{1}{8}-\frac{1}{32}+\cdots $$

Step-by-Step Solution

Answer

The series converges, and its sum is \(\frac{8}{5}\).

Step 1: Identify the first term (a)

The first term of the series is the initial value. Here, the first term is \(a = 2\).

Step 2: Determine the common ratio (r)

Find the common ratio \(r\) by dividing the second term by the first term: \(r = \frac{-\frac{1}{2}}{2} = -\frac{1}{4}\).

Step 3: Check for convergence

A geometric series converges if the absolute value of the common ratio is less than 1: \(\left|r\right| = \left|-\frac{1}{4}\right| = \frac{1}{4} < 1\). Therefore, the series converges.

Step 4: Calculate the sum of the convergent series

The sum \(S\) of an infinite convergent geometric series can be calculated using the formula: \[ S = \frac{a}{1 - r} \] Substitute \(a = 2\) and \(r = -\frac{1}{4}\) into the formula: \[ S = \frac{2}{1 - (-\frac{1}{4})} = \frac{2}{1 + \frac{1}{4}} = \frac{2}{\frac{5}{4}} = \frac{2 \cdot 4}{5} = \frac{8}{5} \]

Key Concepts

convergence

In mathematical terms, convergence means that an infinite series approaches a specific value as more and more terms are added. For a geometric series, convergence is determined by its common ratio, denoted as \(r\). If the absolute value of \(r\), written as \(|r|\), is less than 1, the series will converge. This means the terms of the series will get closer and closer to a particular value.

The geometric series from our exercise is:
\[ 2 - \frac{1}{2} + \frac{1}{8} - \frac{1}{32} + \, ... \, \] The common ratio \(r\) is \(-\frac{1}{4}\). Since \(|-\frac{1}{4}| = \frac{1}{4} < 1\), the series converges. Convergence is fundamental because it assures us that the series has a sum, which allows us to find this sum using a specific formula.

common ratio

The common ratio \(r\) in a geometric series is the factor by which each term is multiplied to get the next term. It's calculated by dividing any term by its preceding term. In our exercise, the series is:

2, -\(\frac{1}{2}\), \(\frac{1}{8}\), -\(\frac{1}{32}\), ...

We get the common ratio by dividing the second term by the first term:
\[r = \frac{-\frac{1}{2}}{2} = -\frac{1}{4}\]
This consistent ratio between consecutive terms is what makes the series 'geometric'.

It's crucial to remember that the behavior of the series largely depends on this ratio. If the absolute value of \(r\) is greater than or equal to 1, the series will diverge, meaning it does not approach a specific value.

sum of series

Once we've confirmed a geometric series converges, we can find its sum using the formula:
\[ S = \frac{a}{1 - r} \]
The term \(a\) is the first term of the series, and \(r\) is the common ratio. For the series in our exercise:

First term \(a = 2\)

Common ratio \(r = -\frac{1}{4}\)

Plugging in these values into our formula:
\[ S = \frac{2}{1 - (-\frac{1}{4})} = \frac{2}{1 + \frac{1}{4}} = \frac{2}{\frac{5}{4}} = \frac{8}{5} \]Therefore, the sum of the series is \( \frac{8}{5} \).

This formula allows us to quickly and easily find the sum of any convergent geometric series, as long as we know the first term and the common ratio.