Problem 43 - Precalculus Mathematics for Calculus

Question

Find the sum. $$\sum_{k=1}^{3} \frac{1}{k}$$

Step-by-Step Solution

Answer

The sum is approximately 1.833.

Step 1: Understanding the Problem

We need to find the sum of a series, which is expressed as a summation: \( \sum_{k=1}^{3} \frac{1}{k} \). This indicates we're to sum the expression \( \frac{1}{k} \) as \( k \) takes on the integer values from 1 to 3.

Step 1: Evaluate the Expression for Each k

Substitute \( k = 1 \) into \( \frac{1}{k} \), which gives \( \frac{1}{1} = 1 \). For \( k = 2 \): \( \frac{1}{2} \) gives 0.5. For \( k = 3 \): \( \frac{1}{3} \), which is approximately 0.333.

Step 2: Sum the Values

Add the results from Step 1: \( 1 + 0.5 + 0.333 \approx 1.833 \).

Step 4: Conclusion

The sum of the series \( \sum_{k=1}^{3} \frac{1}{k} \) is approximately 1.833.

Key Concepts

Harmonic Series

The harmonic series is a fascinating and historical concept in mathematics. It is a specific type of series where each term is the reciprocal of an integer. In notation, it is written as \( rac{1}{1} + rac{1}{2} + rac{1}{3} + rac{1}{4} + rac{1}{5} + ext{...} \), continuing indefinitely.

What's interesting about the harmonic series is that despite each term getting progressively smaller, the sum of the series grows infinitely large. This means it does not converge to a specific number, unlike many other series. The harmonic series plays a crucial role in various areas of mathematics and science, including number theory and analysis. It's particularly important for understanding the behavior and properties of series in general.

Understanding harmonic series helps us appreciate why certain infinite processes, like summing infinitely small positive numbers, can still lead to infinity without reaching a finite bound.

Partial Sum

A partial sum is simply a sum of a portion of an infinite series. It involves calculating the sum up to a specific point, like the first several terms, instead of continuing indefinitely. This gives us a "snapshot" or approximation of what the total sum begins to look like.

For the original exercise of the series \( \sum_{k=1}^{3} \frac{1}{k} \), we computed the partial sum with just the first three terms: \( 1 + \frac{1}{2} + \frac{1}{3} \).
This helps us get an estimate of the behavior of the series over these initial terms. Calculating partial sums is useful in understanding how an infinite series behaves in its early stages and can provide insights into whether the series converges or diverges.

In many practical applications, partial sums are used when we need to approximate a value or when only a limited number of terms are computationally feasible.

Series Evaluation

Evaluating a series is the process of finding a sum for either all or part of a sequence of numbers. It involves deciding how far you want to go in the series and then methodically computing or estimating the sum within those bounds.

In our example with the series \( \sum_{k=1}^{3} \frac{1}{k} \), we evaluate it by plugging in the integers from 1 to 3 for \( k \) and then adding the results:

• \( k = 1 \): \( \frac{1}{1} = 1 \)
• \( k = 2 \): \( \frac{1}{2} = 0.5 \)
• \( k = 3 \): \( \frac{1}{3} \approx 0.333 \)

Adding these values yields an approximate sum of 1.833.

Series evaluation is vital in mathematics, especially when dealing with infinite series, as it allows us to understand and manipulate series for practical and theoretical purposes. Being able to accurately evaluate a series helps in fields ranging from physics to computational math, and even in financial calculations involving interest or investments.