Problem 8 - Algebra 2

Question

Find the sum of each infinite geometric series, if it exists. $$\sum_{n=1}^{\infty} 35\left(-\frac{3}{4}\right)^{n-1}$$

Step-by-Step Solution

Answer

The sum is 20.

Step 1: Identify the First Term

The first term of the geometric series is given by the expression when \( n = 1 \). Therefore, the first term, \( a \), is:\[a = 35 \left(-\frac{3}{4}\right)^{1-1} = 35 \times 1 = 35\]

Step 2: Determine the Common Ratio

The common ratio, \( r \), is the factor each term is multiplied by to get to the next term. In this series, the common ratio is:\[r = -\frac{3}{4}\]

Step 3: Check for the Sum Existence Condition

To find the sum of the infinite series, the series must converge. This occurs when the absolute value of the common ratio is less than 1:\[|r| = \left|-\frac{3}{4}\right| = \frac{3}{4} < 1\]Since this condition is satisfied, the series converges and its sum can be calculated.

Step 4: Calculate the Sum of the Infinite Series

The sum \( S \) of an infinite geometric series is calculated using the formula:\[S = \frac{a}{1 - r}\]Substitute the values of \( a \) and \( r \):\[S = \frac{35}{1 - \left(-\frac{3}{4}\right)} = \frac{35}{1 + \frac{3}{4}}\]Simplify the denominator:\[1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4}\]Calculate the sum:\[S = \frac{35}{\frac{7}{4}} = 35 \times \frac{4}{7} = 20\]

Step 5: Conclusion

The sum of the infinite geometric series is 20.

Key Concepts

Geometric Series Sum

An infinite geometric series is a way to sum up a sequence of terms where each term is obtained by multiplying the previous term by a fixed, constant factor called the common ratio. To find the sum of such a series, a certain condition must be met: the series needs to converge.
For a geometric series with a first term denoted as \( a \) and a common ratio \( r \), the formula to find the sum \( S \) of the geometric series is:

• \( S = \frac{a}{1 - r} \) if \(|r| < 1\)

In the example provided, the series is \( \sum_{n=1}^{\infty} 35\left(-\frac{3}{4}\right)^{n-1} \). To apply the formula for the sum of an infinite geometric series, you need the values for the first term \( a = 35 \) and the common ratio \( r = -\frac{3}{4} \).
If these values meet the conditions for convergence, you can substitute them into the sum formula.

Common Ratio Condition

The common ratio \( r \) is a critical component of a geometric series. It dictates the relation of each term to its predecessor. The nature of this ratio, specifically its absolute value, determines whether a series can have a finite sum, which we refer to as convergence.
For a series to be convergent, the absolute value of the common ratio \( r \) must be less than 1:

• \(|r| < 1\)

In this exercise, the common ratio is \( r = -\frac{3}{4} \). Its absolute value is \( \left| -\frac{3}{4} \right| = \frac{3}{4} \). Since this value is indeed less than 1, the condition for convergence is satisfied, allowing us to calculate the series' sum.

Convergent Series

A convergent series is a series whose terms approach zero as the number of terms grows infinitely large, leading to a finite sum. In simpler terms, even though the series may have infinitely many terms, if it's convergent, they add up to a specific number.
To determine if our given geometric series converges:

• Calculate the absolute value of the common ratio \( r \)
• If \(|r| < 1\), the series converges

For our example, since \( \left| -\frac{3}{4} \right| = \frac{3}{4} < 1 \), the series is convergent. This means we can apply the infinite series formula to find the sum because the terms progressively get closer to zero, ensuring a finite sum emerges.

Infinite Series Formula

Once you have established that a geometric series converges, you can apply the infinite series formula to calculate its sum. This formula simplifies the process of adding infinitely many terms by relying on the first term of the series and the common ratio:

• \( S = \frac{a}{1 - r} \)

For the series \( \sum_{n=1}^{\infty} 35\left(-\frac{3}{4}\right)^{n-1} \), we substitute \( a = 35 \) and \( r = -\frac{3}{4} \) into the formula:

• Denominator becomes \( 1 - (-\frac{3}{4}) = 1 + \frac{3}{4} = \frac{7}{4} \)
• Thus, \( S = \frac{35}{\frac{7}{4}} = 35 \times \frac{4}{7} = 20 \)

The series sums to 20, and this result highlights how the infinite series formula streamlines calculating a series' sum by simply using the common ratio's role in series convergence.