Problem 42 - Essential Calculus

Question

Find the sum of the convergent series. $$ 4-2+1-\frac{1}{2}+\cdots $$

Step-by-Step Solution

Answer

The sum of the series is \(\frac{8}{3}\).

Step 1: Identify the values

First, identify the first term and the ratio of the series. Here, the first term \(a\) is 4 and the ratio \(r\) is \(-\frac{1}{2}\).

Step 2: Apply the geometric series sum formula

Next, apply the formula for the sum of a geometric series, which is \(S = \frac{a}{1-r}\). Substituting in the known values, we get \(S = \frac{4}{1 - (-\frac{1}{2})}\)

Step 3: Calculate the sum

Now, compute this to get the sum. Doing so gives \(S = \frac{4}{1+\frac{1}{2}} = \frac{4}{\frac{3}{2}} = \frac{4*2}{3} = \frac{8}{3}.\)

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This type of series is fascinating because it often approaches a specific value as you sum more and more terms, known as a convergent series.
For example, in the series given in the exercise, which is \(4, -2, 1, -\frac{1}{2}, \ldots \), each term is obtained by multiplying the previous term by \(-\frac{1}{2}\). This makes \(-\frac{1}{2}\) the common ratio. The first term \(a\) is 4, and this series continues indefinitely.
Understanding geometric series is essential because they pop up in various scenarios, from finance where they model interest growth to computer science algorithms analyzing time complexities. Their defining feature, the common ratio, can be positive or negative, leading to interesting behaviors in their sum.

Sum of Series

Calculating the sum of a geometric series helps us understand the overall trend or value the series is approaching. When dealing with an infinite geometric series that converges (meaning the absolute value of the ratio is less than one), you can actually calculate an exact sum.
An infinite geometric series converges when the absolute value of the common ratio \(r\) is less than 1. In these cases, the series approaches a finite limit. For the exercise's series, given \(r = -\frac{1}{2}\), we know it converges since \(|-\frac{1}{2}| < 1\).
This is particularly valuable, as convergent series let us find meaningful totals even when we sum infinitely many terms, a property that is used in fields such as physics and engineering to model continuous systems.

Geometric Series Formula

The geometric series formula is a powerful way to find the sum of a convergent series. For an infinite geometric series with a convergent ratio \( |r| < 1 \), the sum \(S\) is calculated using the formula:

• \( S = \frac{a}{1 - r} \)

Here, \(a\) is the first term of the series, and \(r\) is the common ratio.
Applying this to the provided series, where \(a = 4\) and \(r = -\frac{1}{2}\), we plug these values into the formula:

• \( S = \frac{4}{1 - (-\frac{1}{2})} = \frac{4}{1 + \frac{1}{2}} \)
• \( = \frac{4}{\frac{3}{2}} = \frac{8}{3} \)

This formula is not only useful for finding the sum, but it also reveals how different initial conditions and ratios affect the outcome, demonstrating the critical role of each parameter in determining a series’ behavior.