Problem 14 - Single Variable Calculus

Question

Determine whether the \(p\) -series is convergent or divergent. $$ \sum_{n=1}^{\infty} n^{-0.98} $$

Step-by-Step Solution

Answer

The given \(p\)-series, \(\sum_{n=1}^{\infty} n^{-0.98}\), is divergent because \(p = 0.98 \leq 1\).

Step 1: Identify the p-series and determine the value of p

The given series is a \(p\)-series in the form: $$ \sum_{n=1}^{\infty} n^{-p} $$ Comparing this to the given series, we can identify that the value of \(p = 0.98.\)

Step 2: Apply the p-series test

The \(p\)-series test states that a \(p\)-series converges if \(p > 1\) and diverges if \(p \leq 1\). In this case, we have \(p = 0.98\). Since \(0.98 \leq 1\), the \(p\)-series test states that the given series diverges.

Step 3: Conclusion

The given series, \(\sum_{n=1}^{\infty} n^{-0.98}\), is a divergent \(p\)-series because \(p = 0.98 \leq 1\).

Key Concepts

Series Convergence

In the world of mathematics, series convergence is a crucial concept. A series is a sum of terms of a sequence. We often deal with infinite series, where the number of terms goes to infinity. Convergence of a series means that as we sum more and more terms, the total sum approaches a specific value. This value is known as the limit.
If a series converges, it is well-behaved and leads to a finite value. However, if it doesn't, it diverges, meaning the sum keeps growing indefinitely or doesn't settle on a single value. Understanding whether a series converges or diverges is important for solving problems in calculus, physics, and engineering.
Convergence happens under specific conditions, and there are tests to determine if a series meets these conditions. By understanding the behavior of the series at infinity and using convergence tests, we can grasp whether a series will result in a finite number.

P-Series Test

The p-series test is a simple yet effective tool to determine the convergence of certain series. A p-series takes the form: \[ \sum_{n=1}^{\infty} \frac{1}{n^p} \] The p-series test is particularly useful because it provides a clear criterion to check for convergence or divergence based on the value of \(p\). This test states that:

• If \(p > 1\), the series converges.
• If \(p \leq 1\), the series diverges.

This rule arises from the behavior of the terms of the series as \(n\) becomes very large. When \(p > 1\), the terms decrease quickly enough to add up to a finite sum, while for \(p \leq 1\), they don't, leading to divergence.
Applying the p-series test can save time in determining the behavior of a series without having to calculate sums explicitly.

Divergent Series

A divergent series is a series that does not converge to a limit. This means that as you keep adding more terms, the sum does not approach a particular value; instead, it continues to grow without bound or oscillates without settling.
A classic example of a divergent series is the harmonic series: \[ \sum_{n=1}^{\infty} \frac{1}{n}\] Despite the terms getting smaller, the sum of the series grows indefinitely.
In the given exercise, we looked at a series with \(p = 0.98\). According to the p-series test, because \(p \leq 1\), this series diverges, which means its sum also grows without reaching a stable value.
Understanding divergent series is essential in recognizing the limitations of certain mathematical expressions, and in recognizing when solutions can or can't exist for a given mathematical problem.