Problem 52 - Algebra and Trigonometry

Question

Find the sum of the infinite geometric series. $$ 3-\frac{3}{2}+\frac{3}{4}-\frac{3}{8}+\cdots $$

Step-by-Step Solution

Answer

The sum of the series is 2.

Step 1: Identify the First Term

The first term of the series is the constant term we see first, which in this case is 3.

Step 2: Identify the Common Ratio

The series is geometric, so we find the common ratio by dividing the second term by the first term, \(-\frac{3}{2} \div 3 = -\frac{1}{2}\). This gives us the common ratio \(r = -\frac{1}{2}\).

Step 3: Check for Convergence

A geometric series converges if the absolute value of the common ratio \(|r|<1\). Since \(|-\frac{1}{2}| = \frac{1}{2} < 1\), the series converges.

Step 4: Use the Sum Formula for Convergent Series

The sum of an infinite geometric series can be found using the formula \( S = \frac{a}{1 - r} \), where \(a\) is the first term and \(r\) is the common ratio.

Step 5: Substitute Values into the Formula

Substitute \(a = 3\) and \(r = -\frac{1}{2}\) into the formula to find \( S = \frac{3}{1 - (-\frac{1}{2})} = \frac{3}{\frac{3}{2}} = 2\).

Key Concepts

Geometric Series

In a geometric series, each term is a constant multiple of the previous term. This constant is called the common ratio.
The series typically takes the form: \(a, ar, ar^2, ar^3, \ldots\).
Each term is obtained by multiplying the previous term by the common ratio \(r\).

• The series can be either finite or infinite.
• In our example, the series starts at 3 and each subsequent term is smaller as they are multiplied by the negative common ratio.

Geometric series are very useful because they can model many natural phenomena and financial calculations.

Common Ratio

The common ratio is a key component in determining the nature of a geometric series. It is the factor by which each term of the series is multiplied to get the next term.
To find it, pick any two consecutive terms and divide the latter by the former. In our example, the first term after the initial is \(-\frac{3}{2}\) and dividing it by the first term 3, we find \(-\frac{1}{2}\).
\[ r = \frac{-\frac{3}{2}}{3} = -\frac{1}{2} \]
It's important to note the sign of the common ratio:

• A positive ratio means all terms are in the same direction relative to 0.
• A negative ratio causes terms to alternate in sign.

Understanding the common ratio helps in predicting the behavior of the series.

Convergent Series

A geometric series may either converge or diverge depending on its common ratio.
A series converges if the sum of its infinite terms tends towards a finite value. This occurs when the absolute value of the common ratio is less than one: \(|r| < 1\).
For our series, the common ratio \(-\frac{1}{2}\) meets this criterion because its absolute value is \(\frac{1}{2}\):

• Since this is true, the series converges, meaning the sum of the terms is finite.

Convergent series are crucial in mathematical analysis as they allow us to make predictions about infinite sums.

Sum of Series

To find the sum of an infinite geometric series that converges, we use the formula:
\[S = \frac{a}{1 - r} \]
where \(a\) is the first term of the series and \(r\) is the common ratio.
This formula simplifies the calculation by providing a method to get the total of all terms without adding each manually.
In our exercise, the first term is 3 and the common ratio is \(-\frac{1}{2}\). Substituting these values into the formula gives:

• \[S = \frac{3}{1 - (-\frac{1}{2})} = \frac{3}{\frac{3}{2}} = 2 \]

This tells us that even though the series is infinite, the total value is precisely 2.