Problem 48 - A Graphical Approach to College Algebra

Question

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$2,-10,50,-250, \dots$$

Step-by-Step Solution

Answer

The series does not converge as \(|r| = 5 > 1\).

Step 1: Identify the First Term

The first term of the sequence is given as \( a = 2 \).

Step 2: Find the Common Ratio

The common ratio \( r \) is found by dividing the second term by the first term. Thus, \( r = \frac{-10}{2} = -5 \).

Step 3: Check for Convergence

For an infinite geometric series to converge, the common ratio must satisfy \( |r| < 1 \). Here, \( |r| = |-5| = 5 \), which does not satisfy the condition for convergence.

Key Concepts

Infinite Series

An infinite series is a sum of all infinite terms in a sequence. In contrast to finite series, which have a clear end, infinite series go on indefinitely. When we're dealing with infinite geometric sequences, their series take the special form where each term depends on the one before it, by multiplying by a constant known as the common ratio.
This results in an expression like this:

• First term: 2
• Second term: -10
• And so on, such that the next term is obtained by multiplying by the common ratio

Importantly, not all infinite series will add up to a finite number. This depends largely on the properties of the sequence itself."

Common Ratio

The common ratio in a geometric sequence is the constant factor between consecutive terms. To find it, divide one term by its previous term.
In our sequence:

• The second term is -10
• The first term is 2
• Thus, the common ratio \( r \) is \( \frac{-10}{2} = -5 \)

The common ratio is essential. It dictates how quickly the terms in the sequence grow or shrink. When the common ratio is greater than 1 or less than -1, as in our sequence, terms increase or decrease rapidly.
This significantly affects whether the sum of the sequence converges or diverges."

Convergence

Convergence of a geometric series refers to whether the sum of the series approaches a certain value as more terms are added. For a geometric series to converge, its absolute common ratio must be less than 1, i.e., \(|r| < 1\). This facilitates the terms becoming smaller and the total sum winding down to a specific value.
In the case of our sequence, the common ratio \( r = -5 \) has an absolute value of 5, which doesn't satisfy convergence conditions.
Without convergence, the sum of the series will continue to grow indefinitely and won't settle on a particular number, known as diverging."

Geometric Sequence Formula

The geometric sequence formula gives us a way to find any term in the sequence, or even the sum of the terms in some cases. The general formula for the nth term in a geometric sequence is:\[ a_n = a \cdot r^{n-1} \]where:

• \( a \) is the first term
• \( r \) is the common ratio
• \( n \) is the term number

For infinite series, if the series converges, we also have a handy formula for the sum:\[ S = \frac{a}{1 - r} \]However, this formula only applies when the sequence meets the convergence condition \(|r| < 1\).
Since our sequence does not meet this, its sum cannot be determined using this formula, highlighting the importance of the convergence condition."