Problem 9 - Single Variable Calculus

Question

Determine whether the \(p\) -series is convergent or divergent. $$ \sum_{n=1}^{\infty} \frac{1}{n^{3}} $$

Step-by-Step Solution

Answer

The given \(p\)-series \(\sum_{n=1}^{\infty} \frac{1}{n^3}\) has a \(p\)-value of \(p=3\), which is greater than 1. Therefore, the series is convergent.

Step 1: Identify the \(p\)-value

In the given series \(\sum_{n=1}^{\infty} \frac{1}{n^3}\), we can see that the \(p\)-value is equal to the exponent of \(n\), which is \(p=3\).

Step 2: Compare the \(p\)-value to 1

Now that we have identified the \(p\)-value, we need to compare it to 1. As mentioned in the analysis, if \(p > 1\), the series converges, whereas if \(p \leq 1\), the series diverges. Since \(p=3\), we have \(p > 1\).

Step 3: Determine convergence or divergence

As our comparison in step 2 showed that \(p > 1\), we can conclude that the given \(p\)-series \(\sum_{n=1}^{\infty} \frac{1}{n^3}\) is convergent.

Key Concepts

Infinite Series

An infinite series is a sum of an infinite sequence of terms. It can be expressed as \( \[ \sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \ldots \] \), where \( a_n \) represents the nth term of the series. Infinite series are fascinating mathematical concepts because they extend beyond the finite sums we deal with in everyday life. To determine whether the sum of an infinite series exists (the series converges to a finite value), or does not exist (the series diverges), various convergence tests are applied.

For example, the given exercise deals with the series \( \sum_{n=1}^{\infty} \frac{1}{n^{3}} \), which is an infinite series where each term gets progressively smaller as n increases. Understanding infinite series is crucial because they often appear in physics, engineering, and economics to represent phenomena that are cumulative in nature but whose individual elements diminish over time.

Convergence Tests

Convergence tests are methods to determine if an infinite series converges or diverges. One commonly used test is the p-series test, which evaluates the series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a positive real number. The rule is straightforward: if \( p > 1 \), then the series converges; if \( p \leq 1 \), it diverges. This criterion stems from the integral test, another convergence assessment, which compares the series to the corresponding improper integral.

In our exercise, the p-series test is deployed to analyze the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \), with \( p = 3 \). Since our \( p \) is greater than 1, we quickly conclude that the series converges, based on the p-series test criteria. This kind of test is incredibly valuable, especially when dealing with series that have terms involving powers of n.

Harmonic Series

The harmonic series is a specific type of infinite series and is defined as \( \sum_{n=1}^{\infty} \frac{1}{n} \). It is one of the most famous examples of a divergent series, meaning that its terms do not add up to a finite number. This may seem counterintuitive, since each term in the series is getting smaller, but no finite sum can account for all the terms.

The harmonic series is a benchmark for understanding convergence because its divergence is not immediately apparent. However, it serves as a perfect contrast when learning about p-series; any p-series with \( p \leq 1 \) behaves like the harmonic series and diverges. The exercise provided illustrates a series with \( p = 3 \) which diverges from the harmonic example. Knowing about the harmonic series enriches the grasp of why certain series with \( p > 1 \) converge, highlighting the delicate balance between the size of individual terms and the overall sum's behavior.