Problem 66 - Algebra and Trigonometry

Question

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 4\left(-\frac{1}{2}\right)^{k-1} $$

Step-by-Step Solution

Answer

The series converges, and its sum is \( \frac{8}{3} \).

Step 1: Identify the General Form of the Series

The given series can be written in the general form of a geometric series: the sum of the series can be expressed as the first term, denoted as \(a\), and common ratio, denoted as \(r\).

Step 2: Determine the First Term

Identify the first term (\( a \)) of the series: For the given series: \( 4\bigg(-\frac{1}{2}\bigg)^{k-1} \), the first term is the value of the series when \( k = 1 \): \( a = 4\bigg(-\frac{1}{2}\bigg)^{1-1} = 4 \).

Step 3: Determine the Common Ratio

Identify the common ratio (\( r \)) of the series: The common ratio is the factor by which each term is multiplied to get the next term. For the given series: \( 4\bigg(-\frac{1}{2}\bigg)^{k-1} \): \( r = -\frac{1}{2} \).

Step 4: Check for Convergence

A geometric series converges if the absolute value of the common ratio \( r \) is less than 1. Verify |\( r \)|: \( |-\frac{1}{2}| = \frac{1}{2} < 1 \), so the series converges.

Step 5: Calculate the Sum

Use the formula for the sum of an infinite geometric series: \[ S = \frac{a}{1 - r} \]We know \( a = 4 \) and \( r = -\frac{1}{2} \). Substitute these values into the formula: \[ S = \frac{4}{1 - (-\frac{1}{2})} = \frac{4}{1 + \frac{1}{2}} = \frac{4}{\frac{3}{2}} = \frac{4 \times 2}{3} = \frac{8}{3} \].

Key Concepts

convergence of series

When studying infinite geometric series, one key concept is convergence. A series converges if it approaches a finite value as more and more terms are added. For a geometric series, convergence is determined by the common ratio, denoted as \( r \).
To check if a geometric series converges, examine the absolute value of the common ratio \( |r| \).
The series converges if and only if \( |r| < 1 \).

• For example, in the series \( \sum_{k=1}^{\infty} 4\left(-\frac{1}{2}\right)^{k-1} \), the common ratio \( r \) is \( -\frac{1}{2} \).
• Since \( | -\frac{1}{2} | = \frac{1}{2} \), which is less than 1, the series converges.

In contrast, if \( |r| \geq 1 \), the series diverges, meaning it does not approach a finite value.

sum of infinite series

The sum of an infinite geometric series can be found only if the series converges. For a convergent geometric series, there is a simple formula to calculate the sum.
This formula sums all the terms from the first to infinity, resulting in a finite value.

• The general formula for the sum \( S \) of an infinite geometric series is \[ S = \frac{a}{1 - r} \] where \( a \) is the first term and \( r \) is the common ratio.

Let's apply this to the series \( 4\left(-\frac{1}{2}\right)^{k-1} \):

• First term \( a = 4 \)
• Common ratio \( r = -\frac{1}{2} \)

Substituting into the formula, we get: \[ S = \frac{4}{1 - (-\frac{1}{2})} = \frac{4}{1 + \frac{1}{2}} = \frac{4}{\frac{3}{2}} = \frac{8}{3} \].
So, the sum of the series is \( \frac{8}{3} \).

geometric series formula

A geometric series has a distinct pattern where each term after the first is obtained by multiplying the previous term by a constant, known as the common ratio \( r \).

The general form of a geometric series is:
\sum_{k=1}^{\infty} ar^{k-1}, where:

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

For example, in the series \( 4\left(-\frac{1}{2}\right)^{k-1} \):

• First term \( a = 4 \)
• Common ratio \( r = -\frac{1}{2} \)

Each term in this series is obtained by multiplying the previous term by \( -\frac{1}{2} \).
Understanding this pattern is crucial for identifying and working with geometric series.
This formula allows us to find sums and analyze the behavior of series effectively.