Problem 41 - Algebra and Trigonometry

Question

Find the sum of each infinite geometric series. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots $$

Step-by-Step Solution

Answer

The sum of the infinite geometric series 1 - 1/2 + 1/4 - 1/8 + ... is 2/3.

Step 1: Identify the first term and the common ratio

For the given series 1 - 1/2 + 1/4 - 1/8 + ... , the first term (a) is 1, and the common ratio can be found by dividing the second term by the first term, or the third term by the second term, and so on. Therefore, the common ratio (r) is -1/2.

Step 2: Verify if the absolute value of r is less than one

The absolute value of r (|-1/2|) is 1/2, which is less than 1. This means we can use the formula for the sum of an infinite geometric series.

Step 3: Apply the formula and solve for the sum

The sum of the geometric series (S) can be calculated using the formula: S = a / (1 - r). Substituting the values a = 1 and r = -1/2 into the formula gives: S = 1 / (1 - -1/2) = 1 / (3/2) = 2/3.

Key Concepts

Geometric Series Sum

When we talk about a geometric series, we are referring to a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. Calculating the sum of such a series can be fascinating, especially when the series is infinite. But how do we go about this?

An infinite geometric series can have a sum if the absolute value of the common ratio is less than 1. If it is equal to or greater than 1, the series does not converge and therefore doesn't have a finite sum. The formula to find the sum of an infinite geometric series is given by \( S = \frac{a}{1 - r} \) where \( a \) is the first term and \( r \) is the common ratio. By applying this formula, we can determine that even an infinite number of terms can sometimes add up to a finite amount. This concept might seem counterintuitive, but it's a fundamental aspect of series in mathematics.

Common Ratio

The common ratio in a geometric series is a staple that determines the behavior of the series. It is found by dividing any term in the series by the preceding term (except the first term which has no predecessor). For the infinite series \( 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots \), our common ratio is \( -\frac{1}{2} \).

The common ratio dictates whether the series diverges (spreads out infinitely) or converges (approaches a point). In the case of our example, because the absolute value of our common ratio is less than 1, each term decreases in size, and the series approaches a finite value. It's critical to understand the common ratio because it directly affects whether we can find a sum for the series and what that sum is.

Convergent Series

A convergent series is one where the sum of all its terms approaches a finite limit. This concept is central to understanding how an infinite geometric series can add up to a precise value. The condition for convergence in a geometric series is that the absolute value of the common ratio must be less than one.

In our series example, since the absolute value of the common ratio (\( |-\frac{1}{2}| = \frac{1}{2} \)) is less than 1, the series converges. This is why we can confidently say that the sum of \( 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots \) is \( \frac{2}{3} \), as each succeeding term has less impact on the total sum. Remember that not all infinite geometric series converge; they must meet the specific condition on the common ratio to do so.