Problem 57 - A Graphical Approach to College Algebra

Question

Find each sum that converges. $$\sum_{i=1}^{\infty} 3\left(\frac{1}{4}\right)^{i-1}$$

Step-by-Step Solution

Answer

The sum of the series is 4.

Step 1: Identify the Series Type

The given series is \( \sum_{i=1}^{\infty} 3 \left( \frac{1}{4} \right)^{i-1} \). This is a geometric series of the form \( \sum_{i=1}^{\infty} ar^{i-1} \), where \( a = 3 \) and \( r = \frac{1}{4} \).

Step 2: Check for Convergence

A geometric series \( \sum_{i=1}^{\infty} ar^{i-1} \) converges if the common ratio \( |r| < 1 \). In this case, \( r = \frac{1}{4} \), so \( |r| = \frac{1}{4} < 1 \). Thus, the series converges.

Step 3: Apply the Geometric Series Formula

For a converging geometric series \( \sum_{i=1}^{\infty} ar^{i-1} \), the sum is given by \( S = \frac{a}{1 - r} \). Here, \( a = 3 \) and \( r = \frac{1}{4} \), so the sum is \( S = \frac{3}{1 - \frac{1}{4}} \).

Step 4: Simplify the Expression

Calculate \( 1 - \frac{1}{4} \) which is equal to \( \frac{3}{4} \). Thus, the sum is \( S = \frac{3}{\frac{3}{4}} \).

Step 5: Final Calculation

Simplify the expression \( \frac{3}{\frac{3}{4}} \). This is equivalent to \( 3 \times \frac{4}{3} \), which simplifies to \( 4 \).

Key Concepts

Convergence

Convergence is an essential concept when dealing with infinite series. It helps us determine whether a series sums up to a finite number. In mathematical terms, a series converges if the sequence of partial sums approaches a fixed limit as the number of terms grows indefinitely.
For a geometric series, convergence is all about the common ratio, denoted as \( r \). Specifically, a geometric series \( \sum_{i=1}^{\infty} ar^{i-1} \) will converge if the absolute value of \( r \) is less than 1, that is \( |r| < 1 \).
If \( |r| \) is equal to or greater than 1, the series does not converge and instead diverges, meaning it does not sum to a fixed number. Understanding convergence helps in knowing whether we can find a specific sum for an infinite series.

Geometric Series Formula

The geometric series formula is a powerful tool for calculating the sum of a convergent geometric series. When given a geometric series \( \sum_{i=1}^{\infty} ar^{i-1} \), and knowing it converges, the sum \( S \) can be calculated using the formula:

• \( S = \frac{a}{1 - r} \)

Here, \( a \) is the first term, and \( r \) is the common ratio. This formula assumes \( |r| < 1 \), ensuring the series converges.
Applying this formula is straightforward once you identify \( a \) and \( r \). In our example, with \( a = 3 \) and \( r = \frac{1}{4} \), we calculated the sum to be 4 by substituting into the formula, \( S = \frac{3}{1 - \frac{1}{4}} \). Such calculations can quickly yield the sum of infinitely many terms without actually adding them individually.

Infinite Series

Infinite series are sums of an endless sequence of numbers. They are of significant interest in both mathematics and science. An infinite series is typically denoted as \( \sum_{i=1}^{\infty} a_i \), where each \( a_i \) is a term in the series that extends indefinitely.

Infinite series can be challenging to work with because, by nature, they do not have an obvious finite end. However, geometric series, a specific type of infinite series, can often simplify this complexity. By checking the convergence of the series and understanding the geometric series formula, we can transform the infinite into something finite and manageable.

• Convergence and the geometric series formula allow mathematicians to make accurate predictions and calculations.
• They enable the simplification of complex problems involving ongoing repetitive patterns.

Infinite series play a pivotal role in various fields, from calculating compound interest to predicting behavior in systems modeled by equations.