Problem 62 - A Graphical Approach to Precalculus with Limits

Question

Find each sum. $$\sum_{i=1}^{\infty} 5\left(-\frac{1}{4}\right)^{i-1}$$

Step-by-Step Solution

Answer

The sum is 4.

Step 1: Identify the Geometric Series

The series given is a geometric series. The general form for the sum of an infinite geometric series is \( a + ar + ar^2 + ar^3 + \ldots \). In this question, the first term \( a \) is 5, and the common ratio \( r \) is \(-\frac{1}{4}\).

Step 2: Use the Formula for Infinite Geometric Series

The sum of an infinite geometric series can be found using the formula \( S = \frac{a}{1 - r} \), where \(|r| < 1\). Substitute \( a = 5 \) and \( r = -\frac{1}{4} \) into the formula.

Step 3: Calculate the Sum Using the Formula

Substitute into the formula: \[ S = \frac{5}{1 - \left(-\frac{1}{4}\right)} = \frac{5}{1 + \frac{1}{4}} = \frac{5}{\frac{5}{4}}. \] Simplifying further, \[ S = 5 \times \frac{4}{5} = 4. \]

Step 4: Verify the Convergence Condition

Ensure that \( |r| < 1 \). In this series, \(|-\frac{1}{4}| = \frac{1}{4} < 1\), so the series converges.

Key Concepts

Infinite Series

An infinite series is a sum of an endless sequence of terms. In mathematics, these series continue indefinitely, allowing us to explore interesting patterns and behaviors. Consider an infinite series as if you're adding up numbers that are forever extended. Here's how it works:

• Instead of ending at a particular number of terms, the series continues to grow.
• The infinite series typically starts with a base term and is followed by terms generated using a specific pattern or rule.
• Such series can occasionally be simplified to a single finite value, called the sum of the series.

An example of an infinite series is the geometric series, which grows in a highly structured manner. When dealing with these series, it's crucial to determine whether the series converges or diverges, as understanding this aspect tells us whether the sum is finite or infinite.

Convergence

Convergence is an incredibly important concept when working with infinite series. It refers to the circumstance where the series approaches a specific value as the number of terms increases to infinity.To understand convergence:

• The series converges if adding more terms gets you closer to a fixed sum.
• If the series fails to approach any specific number and continues to grow larger, it diverges.
• For convergence in a geometric series, a condition must be met: the absolute value of the common ratio \(|r|\) must be less than 1.

For example, in the exercise solution, we check the common ratio \(-\frac{1}{4}\). Since \(| -\frac{1}{4} | = \frac{1}{4}\), which is less than 1, we can be certain that the series converges.

Geometric Sequence

A geometric sequence, or geometric series when summed, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio \(r\). This ratio determines the behavior of the series.In a geometric sequence:

• The first term is denoted as \(a\).
• Every subsequent term is obtained by multiplying the preceding term by the common ratio \(r\).
• Written in the form: \(a, ar, ar^2, ar^3, \ldots \)

In the exercise example, the series begins with 5. To find the series' entire sum, the relation between terms via \(r = -\frac{1}{4}\) is crucial. This geometric sequence's characteristic allows us to succinctly calculate its infinite sum using the formula \(S = \frac{a}{1 - r}\) provided \(|r| < 1\). Understanding these series helps you tackle a wide array of mathematical problems efficiently.