Problem 56 - Algebra 2

Question

Find the sum of the two infinite series \(\sum_{n=1}^{\infty}\left(\frac{2}{3}\right)^{n-1}\) and \(\sum_{n=1}^{\infty}\left(\frac{2}{3}\right)^{n}.\)

Step-by-Step Solution

Answer

The sum of the two series is 5.

Step 1: Identify the first term and the common ratio for the first series

The first series is \(\sum_{n=1}^{\infty}\left(\frac{2}{3}\right)^{n-1}\). By substituting \(n = 1\) into the series, we find that the first term 'a' is \(a = \left(\frac{2}{3}\right)^{1-1} = 1\). The common ratio 'r' is given as \(r = \frac{2}{3}\).

Step 2: Calculate the sum of the first series

Using the formula for the sum of an infinite geometric series, \(\frac{a}{1-r}\), substitute the first term and the common ratio found in step 1 to calculate the sum of the first series. The sum is \(S = \frac{1}{1-\frac{2}{3}} = 3.\)

Step 3: Identify the first term and the common ratio for the second series

The second series is \(\sum_{n=1}^{\infty}\left(\frac{2}{3}\right)^{n}\). By substituting \(n = 1\) into the series, we find that the first term 'a' is \(a = \left(\frac{2}{3}\right)^{1} = \frac{2}{3}\). The common ratio 'r' is given as \(r = \frac{2}{3}\).

Step 4: Calculate the sum of the second series

Using the formula for the sum of an infinite geometric series, \(\frac{a}{1-r}\), substitute the first term and the common ratio found in step 3 to calculate the sum of the second series. The sum is \(S = \frac{\frac{2}{3}}{1-\frac{2}{3}} = 2.\)

Step 5: Calculate the sum of the two series

The sum of the two series is the sum of the results from step 2 and step 4, which is \(3 + 2 = 5\).

Key Concepts

Sum of an Infinite Series

The sum of an infinite series is a concept that allows us to find the total of all terms in a sequence that continues forever. When dealing with geometric series, we are often interested in situations where the series gets closer to a certain number. This is made manageable thanks to the properties of geometric series, particularly their common ratio being a fraction less than one. Such series converge to a specific value. To sum an infinite geometric series, we use the formula

• \(S = \frac{a}{1-r}\)

where \(a\) is the first term and \(r\) is the common ratio.
By applying this, we can find the sums for both series in the example, leading us to a total sum of 5.

Common Ratio

The common ratio in a geometric series is a key element. It tells you how each term relates to the previous one. In both series presented, the common ratio \(r\) is \(\frac{2}{3}\). This means each term is \(\frac{2}{3}\) of the previous term.
Understanding this helps you see how the series behaves.

• If \(|r| < 1\), the series will converge.
• If \(|r| \geq 1\), the series will not converge, meaning it does not have a finite sum.

This behavior is crucial for infinite series, helping us determine whether we can even calculate a sum.

Geometric Series Formula

The geometric series formula greatly simplifies the process of summing an infinite series. It is expressed as

• \(S = \frac{a}{1-r}\)

where \(a\) is the first term, and \(r\) is the common ratio. This formula applies only if the series converges, which occurs when \(|r| < 1\).
This formula provides a quick way to sum complex series without calculating each term individually. It was used to find the sums of both series in the example. In these cases, substituting the given values made the process not just straightforward but also efficient.

First Term of a Series

The first term of a geometric series, represented by \(a\), is the starting point from which the series builds. It is essential to correctly identify this term as it is used in the geometric series formula.

• In the first series, \(a = 1\).
• In the second series, \(a = \frac{2}{3}\).

Recognizing the first term may require substituting the value of \(n = 1\) into the general term of the series, just as shown. Accurate identification of \(a\) ensures the subsequent calculations, like finding the sum, are correct.