Problem 67 - A Graphical Approach to Precalculus with Limits

Question

Find each sum. $$\sum_{i=1}^{\infty}\left(\frac{1}{5}\right)\left(-\frac{1}{2}\right)^{i-1}$$

Step-by-Step Solution

Answer

The sum is \( \frac{2}{15} \).

Step 1: Identify the Series Type

The given series is a geometric series with the general term \( a_i = a imes r^{i-1} \), where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = \frac{1}{5} \) and \( r = -\frac{1}{2} \).

Step 2: Check Convergence Criteria

A geometric series converges when the absolute value of the common ratio \( |r| \) is less than 1. In this series, \( |r| = \left|-\frac{1}{2}\right| = \frac{1}{2} < 1 \) indicating that the series converges.

Step 3: Use the Formula for the Sum of an Infinite Geometric Series

The sum of an infinite geometric series is given by the formula \( S = \frac{a}{1 - r} \). Here, \( a = \frac{1}{5} \) and \( r = -\frac{1}{2} \).

Step 4: Calculate the Sum

Substitute the values of \( a \) and \( r \) into the formula: \[ S = \frac{\frac{1}{5}}{1 - \left(-\frac{1}{2}\right)} = \frac{\frac{1}{5}}{1 + \frac{1}{2}} = \frac{\frac{1}{5}}{\frac{3}{2}} = \frac{1}{5} \times \frac{2}{3} = \frac{2}{15} \]. The sum of the series is \( \frac{2}{15} \).

Key Concepts

Convergence of Series

In mathematics, the convergence of a series refers to the condition when a series approaches a specific value as the number of terms increases. For infinite geometric series, we particularly check if the series converges by examining the common ratio, denoted as \( r \).

To determine if a series converges, we look at the absolute value of \( r \). The series converges if the absolute value \( |r| \) is less than 1. This is outlined as a rule:

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

For the given series, the common ratio is \(-\frac{1}{2}\), and the absolute value is \(\frac{1}{2}\). Since \(\frac{1}{2} < 1\), the series converges. This means that as more terms are added to the series, it approaches a specific value instead of growing indefinitely or oscillating.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant known as the common ratio \( r \). This is a consistent pattern observed in geometric sequences.

A general geometric sequence can be represented as \( a_i = a \times r^{i-1} \), where:

• \( a \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( i \) corresponds to the term position in the sequence.

In our problem, the sequence starts with the term \( \frac{1}{5} \), and the common ratio is \( -\frac{1}{2} \). This means each subsequent term is computed by multiplying the preceding term by \(-\frac{1}{2}\). By recognizing that the series is geometric, we can apply specific methods to find the sum of the series.

Sum of Series Formula

When dealing with an infinite geometric series, knowing the sum formula is crucial. Once we affirm a series converges (i.e., \(|r| < 1\)), we can employ the sum formula for the series. It is given as:

\[ S = \frac{a}{1 - r} \]

This formula allows the calculation of the overall sum of the series regardless of the infinite number of terms. Here's a breakdown of this formula and its application:

• \( S \) denotes the sum of the infinite series.
• \( a \) is the first term in the series.
• \( r \) is the common ratio.

In the provided problem, we start with the first term \( a = \frac{1}{5} \), and the common ratio \( r = -\frac{1}{2} \). By substituting these into the formula, the sum is calculated as follows:\[S = \frac{\frac{1}{5}}{1 - (-\frac{1}{2})} = \frac{\frac{1}{5}}{\frac{3}{2}} = \frac{\frac{1}{5} \times \frac{2}{3}}{\frac{1}{1}} = \frac{2}{15}\]
This result means the entire infinite series sums up to \( \frac{2}{15} \). By understanding the sum formula and how to apply it, we can tackle any infinite geometric series confidently.