Problem 14 - 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 " \(r\) ". $$ -2+4-8+16-32+64 $$

Step-by-Step Solution

Answer

The series is not a standard geometric series, but it can be expressed using a modified geometric series rule with \(a = -2\) and \(r = -2\).

Step 1: Identify the first term

The first term of the sequence is \(a = -2.\)

Step 2: Identify if the sequence is geometric

To determine if the sequence is geometric, check if each term is a constant multiple (r) of the previous term. \n Let's compute the ratio between the successive terms: \(4/-2 = -2 = r, -8/4 = 2, 16/-8 = -2, -32/16 = 2, 64/-32 = -2\). We observe that the ratio alternates between -2 and 2, which indicates that this is not a standard geometric series.

Step 3: Formulate the general rule

Although this is not a standard geometric series, we can formulate a modified version of the geometric series formula for this sequence: the nth term will be \(a * (-2)^n \) where \(n\) starts from 0.

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This fixed number can be any real number and determines the characteristic of the sequence. For example, in the sequence 2, 4, 8, 16, ..., each term is 2 times the term before. Hence, it's a geometric sequence with a common ratio of 2.

The general form of a geometric sequence is: \[ a, ar, ar^2, ar^3, \ \text{...} \text{where} \ a \text{is the first term, and} \ r \text{is the common ratio.} \] Understanding geometric sequences is fundamental as they are used to model exponential growth or decay in various applications such as finance, computer science, and natural sciences.

Series Convergence

When dealing with series, particularly geometric series, an important concept is convergence. Series convergence refers to whether the series approaches a finite number when an infinite number of terms are summed up. A geometric series converges if its common ratio has an absolute value less than 1, that is, \( |r| < 1 \). In such cases, you can find the sum of the series using the formula:\[ S = \frac{a}{1 - r} \]where \( S \) is the sum of the series, \( a \) is the first term, and \( r \) is the common ratio.

For a geometric series to converge, the terms should get progressively smaller and approach zero. If the ratio’s absolute value is greater than or equal to 1, the terms will not approach zero; such a series is called divergent because its sum is infinite or not defined.

Ratio of Successive Terms

The ratio of successive terms in a sequence is a crucial concept for identifying whether a sequence is geometric. To determine whether a sequence is geometric, you can divide any term in the sequence (after the first) by the term immediately preceding it; this should give the same value for every term if the sequence is truly geometric. This consistent value is the common ratio \( r \). In our exercise example, the ratio between successive terms alternates between -2 and 2:\( \frac{4}{-2} = -2, \frac{-8}{4} = -2, \frac{16}{-8} = -2, \) etc., which might suggest that the sequence is not geometric. However, considering that the signs alternate between positive and negative, it reveals an underlying pattern that involves a power of -1. This indicates that a modified version of the geometric sequence formula can be applied, which involves alternating signs, typically by including a \( (-1)^n \) term in the formula for the nth term.