Question

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 8+4+2+\cdots $$

Short answer

The series converges and its sum is 16.

Step-by-step solution

Identify the first term and common ratio

In a geometric series, the first term is denoted as \(a\) and the common ratio is denoted as \(r\). For the series \( 8 + 4 + 2 + \cdots \), the first term \(a\) is 8. To find the common ratio, divide the second term by the first term: \( r = \frac{4}{8} = \frac{1}{2} \).

Determine if the series converges or diverges

A geometric series converges if the absolute value of the common ratio \(|r| < 1\). In this case, \( |\frac{1}{2}| = \frac{1}{2} < 1 \). Thus, the series converges.

Find the sum of the convergent series

The sum \(S\) of an infinite convergent geometric series is given by the formula \( S = \frac{a}{1 - r} \). Substituting the values \(a = 8\) and \(r = \frac{1}{2}\) into the formula, we get: \[ S = \frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} = 8 \times 2 = 16 \]. So, the sum of the series is 16.

Key concepts

convergent series

An infinite geometric series is a sum of an infinite number of terms. Each term is obtained by multiplying the previous term by a constant called the common ratio. When dealing with infinite geometric series, an important concept is whether the series converges or diverges.
A series is said to be convergent if the sum of its terms approaches a finite number as more terms are added. To determine this, we use the absolute value of the common ratio, denoted as \( |r| \). If \( |r| < 1 \), the series converges, meaning it sums to a specific, finite value. If \( |r| \) is equal to or greater than 1, the series diverges, and its sum is infinite or undefined. In the given exercise, the common ratio \( r \) is \( \frac{1}{2} \). Since \( |r| = \frac{1}{2} < 1 \), the series converges.

common ratio

The common ratio is a key element in understanding geometric series. It is the factor by which each term is multiplied to get to the next term. Knowing the common ratio helps us determine various properties of the series, such as whether it converges or diverges.
In the given series \[ 8 + 4 + 2 + \cdots \], the common ratio \( r \) is found by dividing the second term by the first term: \( r = \frac{4}{8} = \frac{1}{2} \). Every term in the series is half of the term before it.
To summarize:

• If \( |r| < 1 \), the series converges.
• If \( |r| \geq 1 \), the series diverges.

The value of the common ratio helps determine the behavior of the entire series.

sum formula

The sum of an infinite convergent geometric series is calculated with a specific formula. This formula only works if the series is convergent (i.e., \( |r| < 1 \)).
The formula for the sum \( S \) of an infinite geometric series is:\br\br \[ S = \frac{a}{1 - r} \]
\brHere, \( a \) is the first term, and \( r \) is the common ratio. In the provided exercise, the first term \( a \) is 8, and the common ratio \( r \) is \( \frac{1}{2} \). Substituting these into the formula gives us:
\br \[ S = \frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} = 8 \times 2 = 16 \]
\brThus, the sum of the infinite geometric series \[ 8 + 4 + 2 + \cdots \] is 16.