Problem 14 - Applied Calculus

Question

Find the sum, if it exists. $$ -2+1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\cdots $$

Step-by-Step Solution

Answer

The sum of the series is \(-\frac{4}{3}\).

Step 1: Identify the Type of Series

The series is \(-2 + 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{16} - \cdots\). This series is an infinite geometric series with the first term \(a = -2\) and a common ratio \(r = -\frac{1}{2}\).

Step 2: Check Condition for Geometric Series Convergence

For an infinite geometric series \(a + ar + ar^2 + \cdots\) to converge, the common ratio \(r\) must satisfy \(|r| < 1\). In this series, \(r = -\frac{1}{2}\), and \(|r| = \frac{1}{2} < 1\), so the series converges.

Step 3: Calculate the Sum of the Geometric Series

The sum \(S\) of an infinite geometric series is given by the formula \(S = \frac{a}{1 - r}\). Substitute \(a = -2\) and \(r = -\frac{1}{2}\) into the formula:\[S = \frac{-2}{1 - (-\frac{1}{2})} = \frac{-2}{1 + \frac{1}{2}} = \frac{-2}{\frac{3}{2}}\]Simplifying further:\[S = -2 \cdot \frac{2}{3} = -\frac{4}{3}\].

Key Concepts

Geometric Series Convergence

In mathematics, an infinite geometric series is a series of the form \(a + ar + ar^2 + ar^3 + \cdots\). Understanding whether or not this series converges is crucial. Convergence means the series adds up to a definite number.
For a geometric series to converge, the absolute value of its common ratio must be less than 1. Mathematically, this condition is expressed as \(|r| < 1\). In the provided series, the common ratio is \(-\frac{1}{2}\), and we find \(| -\frac{1}{2} | = \frac{1}{2}\), indeed less than 1.
In simple terms, this condition ensures that each successive term in the series becomes smaller and closer to zero as the series progresses. Consequently, the series reaches a point where further terms add little to the overall sum, making it converge to a specific value.

Common Ratio

The common ratio in a geometric series is a key component. It is the factor by which each term in the series is multiplied to get the next term. In formula terms, if you take the second term and divide it by the first term, you get the common ratio: \(r = \frac{a_2}{a_1}\).
For our series: \(-2, 1, -\frac{1}{2}, \cdots\), the common ratio \(r\) is determined by: \(r = \frac{1}{-2} = -\frac{1}{2}\). This means each term is obtained by multiplying the previous one by \(-\frac{1}{2}\).
Understanding the common ratio helps predict the behavior of the series. If the ratio is outside the range where the series converges, the series may not reach a finite value.

Sum of an Infinite Series

If an infinite geometric series converges, it is possible to calculate its sum using a simple formula. The sum \(S\) of a converging infinite geometric series is \(S = \frac{a}{1 - r}\), where \(a\) is the first term and \(r\) is the common ratio.
By substituting the values from the problem, where the first term \(a = -2\) and the common ratio \(r = -\frac{1}{2}\), this formula becomes:\[S = \frac{-2}{1 - (-\frac{1}{2})} = \frac{-2}{1 + \frac{1}{2}} = \frac{-2}{\frac{3}{2}}\]
After simplifying, we find:\[S = -2 \cdot \frac{2}{3} = -\frac{4}{3}\]
This result is the finite sum to which our infinite series converges. Even an "infinite" sum can become a manageable finite number when conditions are just right, thanks to convergence.