Problem 114 - Precalculus with Limits

Question

In Exercises 113-116, find the indicated partial sum of the series. \( \displaystyle \sum_{i=1}^{\infty} 2 \left(\dfrac{1}{3} \right)^i \) Fifth partial sum

Step-by-Step Solution

Answer

The fifth partial sum of the series is \( \displaystyle S_5 = \dfrac{2}{3} \left(1 - \left(\dfrac{1}{3}\right)^5\right) / \left(1 - \dfrac{1}{3}\right) \).

Step 1: Identify the First Term and Common Ratio

First, let's identify the first term (a) and the common ratio (r) from the series. The first term 'a' can be calculated by substituting i=1 in the series, which is 2*(1/3)^1 = 2/3. The common ratio 'r' is (1/3).

Step 2: Determine the Number of Terms

The problem asks for the fifth partial sum, which means we are considering the first 5 terms of the series. So, n = 5.

Step 3: Calculate the Partial Sum

Use the formula for the nth partial sum of a geometric series, which is given by: S_n = a*(1 - r^n)/(1 - r). Substituting a = 2/3, r = 1/3, and n = 5 into this formula gives: S_5 = (2/3)*(1 - (1/3)^5)/(1 - 1/3). Calculate the expressions in the parentheses first, then perform the multiplication and division to find the value of S_5.

Key Concepts

Geometric Series

A geometric series is a sum of terms in which each term is a constant multiple of the previous one. This constant is known as the common ratio. Such series can be finite or infinite. For example, in the series \( a, ar, ar^2, ar^3, \ldots \), each term is obtained by multiplying the previous term by \( r \), the common ratio.
Understanding this type of series is crucial because many real-world phenomena, like interest calculations and population growth models, can be described using them. When working with a geometric series, identifying the first term and common ratio is essential so that you can apply formulas to find sums or other properties.
Geometric series can either converge or diverge depending on the absolute value of the common ratio. This determines whether the series approaches a finite limit or not.

Common Ratio

The common ratio \( r \) in a geometric series is the factor by which we multiply to get from one term to the next. In the example given, \( r = \frac{1}{3} \).
Identifying the common ratio quickly allows you to predict and calculate subsequent terms in the series. To find \( r \), simply divide any term in the series by its preceding term.

• If \( |r| < 1 \), the series will converge, meaning it adds up to a finite number as you include more terms.
• If \( |r| \geq 1 \), the series diverges, meaning the sum will not settle on a single value as more terms are added.

Knowing the value of \( r \) helps in determining whether a series will have a finite sum or not, which is crucial when calculating partial sums or exploring infinite series.

Formula for nth Partial Sum

To find the partial sum \( S_n \) of the first \( n \) terms of a geometric series, use the formula:\[ S_n = a \frac{1 - r^n}{1 - r} \]In this formula, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
Consider the series given: its first term \( a = \frac{2}{3} \), common ratio \( r = \frac{1}{3} \), and \( n = 5 \) for the fifth partial sum. Substituting these values gives:\[ S_5 = \left(\frac{2}{3}\right) \frac{1 - \left(\frac{1}{3}\right)^5}{1 - \frac{1}{3}} \]
This formula lets you calculate the sum without actually listing and adding up all the terms, which is both efficient and less error-prone. Employing this method, especially for large \( n \), simplifies calculations extensively.

Infinite Series

An infinite series is the sum of an infinite sequence of terms. Infinite geometric series occur when there are endless terms to add. Understanding if an infinite series converges or diverges is key.

• For an infinite geometric series to converge, the common ratio \( r \) must satisfy \( |r| < 1 \).
• If it converges, the infinite sum \( S \) is given by:\[ S = \frac{a}{1 - r} \]
• If \( |r| \geq 1 \), the series will diverge, meaning its sum is not finite.

This concept is particularly important in mathematical analysis and calculus, where it aids in determining the behavior of functions and models over time. Understanding how infinite series work allows you to grasp the concept of limits and convergence more broadly.