Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. \(\sum_{m=1}^{\infty} 4^{m-1}\)

Step-by-Step Solution

Answer

The series \( \sum_{m=1}^{\infty} 4^{m-1} \) does not have a sum because it diverges.

Step 1: Identify the Series Type

The given series is \( \sum_{m=1}^{\infty} 4^{m-1} \). Recognize that this is an infinite geometric series because it has a constant ratio of consecutive terms.

Step 2: Determine the Common Ratio

To find the common ratio \( r \), look at the form of the series. For a geometric series, \( a_m = a \, r^{m-1} \) where \( a \) is the first term and \( r \) is the common ratio. Here, each term is \( 4^{m-1} \), so the common ratio, \( r = 4 \).

Step 3: Check Convergence Condition

A geometric series converges if the absolute value of the common ratio is less than 1 (\( |r| < 1 \)). In this case, \( |r| = 4 \), which is greater than 1.

Step 4: Conclusion on Convergence

Since the common ratio \( r = 4 \) is greater than 1, the series does not converge. Therefore, the infinite series does not have a sum.

Key Concepts

Geometric Series

A geometric series 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. Geometric series can be either finite or infinite. This concept is crucial in understanding how values can grow exponentially in a sequence.

• In a finite geometric series, we have a limited number of terms.
• An infinite geometric series, as the name implies, has terms that continue indefinitely.

Geometric series have broad applications in finance, science, and many other fields, where understanding exponential growth or decay is essential.

Common Ratio

The common ratio is a key feature that defines a geometric series. It is the factor by which the terms of the series increase or decrease. In the series presented above, the common ratio is determined by observing the consistent multiplication between consecutive terms.

• For the series \( \sum_{m=1}^{\infty} 4^{m-1} \), each term is the previous term multiplied by 4.
• This constant factor, 4, is the common ratio \( r \).

Knowing the common ratio is crucial for determining the behavior of a series, such as whether it converges or diverges.

Series Convergence

Series convergence refers to the idea that the sum of an infinite series approaches a specific value as more and more terms are added. This concept is vital when dealing with infinite series, particularly in distinguishing between series that converge to a finite sum and those that do not.

• A geometric series will converge if the absolute value of its common ratio is less than 1, i.e., \(|r| < 1\).
• For our example, since the common ratio \( r = 4 \) and \(|4| > 1\), the series diverges.

Thus, the series \( \sum_{m=1}^{\infty} 4^{m-1} \) does not converge, meaning it will not sum to a finite number as the number of terms increases.

Sum Formula

The sum formula for a geometric series allows us to calculate the sum of its terms. For an infinite geometric series to have a sum, it must converge. This is why understanding convergence is crucial before applying the sum formula.

• When \(|r| < 1\), the sum \( S \) of an infinite geometric series can be calculated using the formula \( S = \frac{a}{1-r} \), where \( a \) is the first term of the series.
• In the given series, since \(|r| = 4\) which is greater than 1, the sum formula cannot be applied.

Without convergence, calculating a sum for the series is not possible, as it doesn’t reach a finite value.