Problem 6 - 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 " \(r\) ". $$ 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{100} $$

Step-by-Step Solution

Answer

The given series is not a geometric series because the ratio between successive terms is not constant.

Step 1: Identify the first term and common ratio

In a potential geometric series, the first term is 1 and each subsequent term appears to be the result of dividing the previous term by one plus the denominator of the previous term.

Step 2: Verify common ratio

Taking the second term and dividing it by the first term, we get \( \frac{1/2}{1} = 1/2 \). Now, take the third term and divide it by the second term, we get \( \frac{1/3}{1/2} = 2/3 \). For a series to be geometric, the ratio \( r \) would have to be consistent across all successive terms. However, \( 1/2 \) is not equal to \( 2/3 \). So, these are not the same, hence it is not a geometric series.

Step 3: Conclude the result

Since the ratio is not consistent, it can be stated that this series is not a geometric series.

Key Concepts

First Term

In a geometric series, the first term is crucial because it sets the stage for the entire sequence. It is often denoted by the letter \( a \). In this exercise, the first term is \( 1 \). This term is the starting point and from it, every subsequent term is generated by multiplying with the common ratio.

Understanding the first term helps in depicting how subsequent numbers grow or shrink in a geometric series. In this particular problem, identifying the first term allows us to attempt to identify the series and analyze its behavior.

Common Ratio

A common ratio in a geometric series is the factor by which we multiply each term to get the next one. It is denoted as \( r \). Finding a consistent common ratio throughout the series indicates that we are indeed dealing with a geometric series.

In the provided series example, trying to divide consecutive terms, such as the second term (\( \frac{1}{2} \)) by the first term (\( 1 \)), gives us a different ratio than dividing the third term by the second. This inconsistency (\( \frac{1/2}{1} = 1/2 \) compared to \( \frac{1/3}{1/2} = 2/3 \)) shows that there is no single common ratio.

This is why the series presented is not geometric. Each term needs to have the same multiplication factor to qualify as a geometric sequence.

Sequence Pattern

Establishing a sequence pattern is vital to understanding the nature of any series. In a geometric series, the pattern can quickly be identified by looking for a consistent multiplier, or ratio, between each term. If this pattern is disrupted, as it is in this exercise, then the series cannot be classified as geometric.

For visually recognizing patterns:

• It helps to write out the first few terms.
• Calculate the ratio between each term.
• Verify if this ratio remains unchanged across additional terms.

For this series, the pattern doesn't adhere to the criteria of a geometric sequence. This makes it a simple list of reciprocals increasing by one in the denominator without a predictable multiplier.

Sum Determination

Determining the sum of a geometric series involves using a specific formula, which holds true only for geometric sequences. This formula is

\[ S_n = a \frac{1 - r^n}{1 - r} \]

where \( S_n \) stands for the sum of the first 'n' terms, \( a \) is the first term, \( r \) is the common ratio and \( n \) is the number of terms you want to sum.

Unfortunately, since the sequence given is not a geometric one, this formula would not correctly determine its sum. For non-geometric series like this one, each term needs to be added individually as is, without any shortcut formula. This provides a good reminder that sum formulas for geometric series are only applicable when the sequence genuinely follows the geometric pattern.