Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ -2+4-6+8-10+\cdots+16 $$

Step-by-Step Solution

Answer

Yes, the series is geometric. The first term \(a\) is -2 and the common ratio \(r\) is -2.

Step 1: Identify the pattern

The given series is -2, 4, -6, 8, -10, ..., 16. It seems that the pattern of the series is that each term is 2 units greater than the term before it, but the sign of each term alternates between negative and positive.

Step 2: Check if the series is geometric

To determine if a series is geometric, we must verify if there is a common ratio \(r\) by which each term can be multiplied to give the next term. It looks like if we multiply each term by -2, we get the next term. For example, -2*(-2) = 4, 4*(-2) = -8, -8*(-2) = 16 which matches our series.

Step 3: Identify \(a\) and \(r\)

If the series is geometric and has the common ratio we have identified, then we can say that the first term \(a\) (also known as the initial value) is -2 and the common ratio \(r\) is -2.

Key Concepts

Common Ratio

In the world of sequences, particularly geometric ones, the 'common ratio' is the central character. Our investigation into whether a sum is geometric or not hinges on finding this ratio, represented by the letter 'r'. It is the consistent factor that, when applied to one term, gives us the next term in the series.

How does one identify this elusive 'r'? Simply take any term in the series (after the first) and divide it by the term directly preceding it. If this value stays constant throughout the series, you've found your common ratio! For instance, in the series -2, 4, -8, 16,... we divide 4 by -2, obtaining -2. That's our cue to check the next pair: -8 divided by 4, which also equals -2. This consistent -2 confirms that the series indeed has a common ratio and thus, is a geometric series.

Series Convergence

When a mathematician talks about 'series convergence', they're exploring the narrative of whether a series marches towards a finite limit, or runs off into the vastness of infinity. Convergence is essential to understanding the behavior of series over long periods—or into infinity itself!

For a geometric series, convergence is a dance of numbers: if the common ratio's absolute value is less than 1, the series converges; above 1, and the series diverges, wandering infinitely without a destination. Our geometric series, with a common ratio of -2, heads off into infinity because the absolute value of -2 is greater than 1. This means that we cannot find a sum of the series in the classical sense—it doesn’t settle on a finite number.

Arithmetic Series

Now, let us pivot to a different sequence altogether—the arithmetic series. Unlike its geometric cousin, an arithmetic series thrives on a consistent 'common difference' between consecutive terms, rather than a ratio. Think of it like a staircase with equal steps: each term is simply the previous term plus (or minus) a fixed number.

In practice, identify an arithmetic series by checking the difference between terms. If the series -2, 4, -6, 8, -10, ..., 16, were arithmetic, you'd expect each step to have the same 'height'. However, the inconsistency in difference due to alternating signs, swiftly debunks the arithmetic theory for this series. A true arithmetic series has a linear, predictable growth or decay, which our alternating-sign series lacks.