Problem 43 - College Algebra

Question

Find the sum of each infinite geometric series. $$\sum_{i=1}^{\infty} 8(-0.3)^{i-1}$$

Step-by-Step Solution

Answer

The sum of the infinite geometric series \(\sum_{i=1}^{\infty} 8(-0.3)^{i-1}\) is approximately \(6.153846154\).

Step 1: Identify the first term and the common ratio

The first term in the series, denoted as \(a\), is 8. The common ratio, denoted by \(r\), is -0.3. It is important to ensure that the absolute value of the common ratio is less than 1 for the series to be convergent and for the formula for the sum to hold true. Here, |\(-0.3\)| < 1, so the series is convergent.

Step 2: Apply the formula for the sum of an infinite geometric series

Now that we've identified the first term (\(a = 8\)) and the common ratio (\(r = -0.3\)), we can apply the formula for the sum of an infinite geometric series. This gives us \(S = \frac{a}{1-r}\), which simplifies to \(S = \frac{8}{1-(-0.3)}\).

Step 3: Perform the calculations

Carry out the calculation in the numerator and the denominator separately. This simplifies to \(S = \frac{8}{1.3}\). Next, perform the final calculation to determine the value of \(S\).

Key Concepts

First Term

The first term of an infinite geometric series is the initial value from which the sequence starts. In our given series, the first term \(a\) is 8.
Understanding the first term is crucial because it sets the pace for the entire series.
When you look at the series formula, \(\sum_{i=1}^{\infty} 8(-0.3)^{i-1}\), the number 8 does not change, regardless of how many terms you calculate.

• This number represents the starting point of your series.
• It is essential when using the sum formula for infinite series.

Every term that follows in the series is a multiple of the first term, modified by the common ratio. Therefore, always start by identifying the first term to understand the subsequent terms that follow.

Common Ratio

The common ratio is the factor by which we multiply each term of a geometric series to get the next term. For the series \(\sum_{i=1}^{\infty} 8(-0.3)^{i-1}\), the common ratio \(r\) is \(-0.3\).
This value determines how quickly the series terms decrease or increase in magnitude.

• If the absolute value of \(r\) is less than 1, the series terms will decrease, causing the series to converge.
• If \(|r|\) is greater than or equal to 1, the series will not converge, and the sum would not be finite.

Because \(|-0.3| = 0.3\), which is less than 1, this ensures that the terms are decreasing in magnitude.
Thus, the series is convergent, which brings us to the next concept.

Convergent Series

A convergent series is a series where the sum of the terms approaches a finite number as more terms are added. To determine if an infinite geometric series is convergent, check the absolute value of the common ratio.
In our example, \(|-0.3| = 0.3\), which is less than 1, confirming the series is convergent.

• This means that as we continue adding terms, they grow smaller and eventually have minimal impact on the total.
• The formula \(S = \frac{a}{1-r}\) can then be used to find the sum, where \(S\) is the sum of the infinite series.

In this case, substituting \(a = 8\) and \(r = -0.3\) into the formula gives \(S = \frac{8}{1 - (-0.3)} = \frac{8}{1.3}\).
Performing the calculation will provide the finite sum of this convergent series.