Problem 3 - Essential Calculus Early Transcendentals

Question

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? $$\sum_{n=1}^{\infty} \frac{1}{n^{3}}$$

Step-by-Step Solution

Answer

The series appears to be convergent as the partial sums approach a limit.

Step 1: Understand the Sequence of Partial Sums

The sequence of partial sums for an infinite series is given by \( S_N = \sum_{n=1}^{N} \frac{1}{n^3} \). Our task is to calculate \( S_1, S_2, \ldots, S_8 \).

Step 2: Calculate the First Partial Sum

Compute \( S_1 = \frac{1}{1^3} = 1.0000 \).

Step 3: Calculate the Second Partial Sum

Compute \( S_2 = \frac{1}{1^3} + \frac{1}{2^3} = 1 + \frac{1}{8} = 1.1250 \).

Step 4: Calculate the Third Partial Sum

Compute \( S_3 = S_2 + \frac{1}{3^3} = 1.1250 + \frac{1}{27} \approx 1.1620 \).

Step 5: Calculate the Fourth Partial Sum

Compute \( S_4 = S_3 + \frac{1}{4^3} = 1.1620 + \frac{1}{64} \approx 1.1785 \).

Step 6: Calculate the Fifth Partial Sum

Compute \( S_5 = S_4 + \frac{1}{5^3} = 1.1785 + \frac{1}{125} \approx 1.1865 \).

Step 7: Calculate the Sixth Partial Sum

Compute \( S_6 = S_5 + \frac{1}{6^3} = 1.1865 + \frac{1}{216} \approx 1.1911 \).

Step 8: Calculate the Seventh Partial Sum

Compute \( S_7 = S_6 + \frac{1}{7^3} = 1.1911 + \frac{1}{343} \approx 1.1939 \).

Step 9: Calculate the Eighth Partial Sum

Compute \( S_8 = S_7 + \frac{1}{8^3} = 1.1939 + \frac{1}{512} \approx 1.1959 \).

Step 10: Analyze Convergence

The partial sums \( S_n \) are increasing and approaching a limit, suggesting convergence. The changes between subsequent sums decrease, which is characteristic of a convergent series.

Key Concepts

Sequence of Partial Sums

When dealing with infinite series, most students first encounter the concept of the sequence of partial sums. This sequence is formed by successively adding terms of the series and observing how these sums behave. Imagine, for example, that someone gives you an infinite sequence of numbers. Instead of taking them all at once (an impossible task!), you gather them piece by piece.

In mathematical terms, given the infinite series \( \sum_{n=1}^{\infty} a_n \), the sequence of partial sums is denoted as \( S_N \) where \( S_N = \sum_{n=1}^{N} a_n \). Essentially, \( S_1 \) is just the first term \( a_1 \), \( S_2 \) is the sum of the first two terms, and so on. This sequential sum allows us to "build up" towards the whole series.

To check how this works, let's look at an example where \( a_n = \frac{1}{n^3} \). If you calculate each \( S_n \) up to \( n = 8 \) (as demonstrated in the solution), you notice how each additional term contributes less than the one before. Thus, observing this sequence of partial sums can give us insight into whether the series approaches a finite limit.

Convergence

A key inquiry when exploring infinite series is whether they converge. Convergence means that as you keep adding more elements, the sequence of partial sums \( S_N \) approaches a specific value. Conversely, if no limit is approached, the series diverges.

In practical terms, you can observe this by looking at the calculated partial sums. If each subsequent term contributes less and less to the total and becomes negligible, then the series is likely convergent. Take our example, where \( a_n = \frac{1}{n^3} \). The partial sums you've calculated suggest a pattern: each new term's addition results in a smaller increment than the last.

Understanding convergence is crucial because it assures stability. For any applications involving this series, knowing that it converges helps predict behavior accurately, as the sum approaches a reliable and finite value as \( n \to \infty \).

Mathematical Series

The world of mathematical series is vast and fascinating, offering both challenges and exciting insights. A series is essentially the sum of a sequence of numbers, where the sequence can be finite or infinite. In simpler terms, it's like asking "What do we get if we add up all these numbers?"

There are many famous series in mathematics, such as geometric and harmonic series. Our current focus, however, involves infinite series like \( \sum_{n=1}^{\infty} \frac{1}{n^3} \). This particular series is known as a p-series with \( p = 3 \). A strong concept in mathematics is understanding which series converge based on the properties of their terms.

Approaching series requires patience and practice, especially when dealing with infinite ones. It's not just about punching numbers into a calculator, but really understanding the patterns that emerge from the sums. When you grasp the concept of series, it opens up whole new levels of mathematical reasoning and problem-solving.