Problem 18 - Single Variable Calculus

Question

Determine whether the given series is convergent or divergent. $$ \sum_{n=1}^{\infty} n^{-0.75} $$

Step-by-Step Solution

Answer

The given series is \(\sum_{n=1}^{\infty} n^{-0.75}\), which is in the form of a p-series with \(p=0.75\). Using the p-test, the series converges if \(p > 1\) and diverges if \(p \leq 1\). Since \(0.75 \leq 1\), the series is divergent.

Step 1: Identify the p-value

The given series is of the form \(\sum_{n=1}^{\infty} n^{-p}\) with \(p = 0.75\).

Step 2: Apply the p-test

Using the p-test, we will now check if the p-value satisfies the condition for convergence. In this case, we need to check if \(p > 1\).

Step 3: Compare p-value to 1

Since we have \(p = 0.75\), it is clear that \(0.75 \leq 1\).

Step 4: Determine if the series is convergent or divergent

As per the p-test, since \(0.75 \leq 1\), the given series \(\sum_{n=1}^{\infty} n^{-0.75}\) is divergent.

Key Concepts

Series Convergence

A core concept in understanding infinite series is series convergence. Convergence refers to whether the terms of a series approach a specific value as more terms are added. When a series converges, it means there is a finite limit that the sum approaches. Convergent series play a crucial role in mathematics because they allow us to make precise calculations and predictions in various fields.

• Checking for convergence requires examining the behavior of terms as the number of terms grows indefinitely.
• Tests for convergence, such as the p-series test, help in determining whether a series is convergent or divergent.

Understanding series convergence helps in many applications, from engineering to economics, where infinite processes or sums need to be considered carefully.

Mathematical Series

A mathematical series is the sum of the terms of a sequence. For instance, the series given in the exercise is \(\sum_{n=1}^{\infty} n^{-0.75}\), which means you add up the terms \(n^{-0.75}\) starting from \(n = 1\) to infinity.

• Each term in a series is derived based on a mathematical rule or function.
• The notation \(\sum\) indicates summation, which involves adding all the terms together.
• Some series are simple arithmetic or geometric series, while others, like the one given, follow a rule expressed with exponents or other functions.

Series can be finite, with a set number of terms, or infinite, continuing indefinitely. Studying series is key to understanding more complex mathematical concepts.

Divergence in Series

Divergence in a series indicates that as more terms are added, the series does not settle towards a specific value. Instead, it continues to grow larger or fluctuates without bound. In this exercise, the series \(\sum_{n=1}^{\infty} n^{-0.75}\) demonstrates divergence.

• When a series is divergent, it implies that its sum could potentially be infinite or not meaningful in the way convergent sums are.
• The p-series test is often used to test for divergence where certain conditions indicate if a series will diverge or converge.
• Applying the p-series test here, because \(p = 0.75\) is less than 1, it shows the series is divergent.

Understanding divergence helps mathematicians know when a series does not have a useful sum for practical calculations.

P-value Analysis

P-value analysis in relation to series involves determining the value of \(p\) in a p-series. The p-value influences whether a series converges or diverges based on specific conditions outlined by the p-series test. Here, \(p = 0.75\) was analyzed to determine the nature of the series.

• If \(p > 1\), the p-series converges, and there’s a finite sum.
• If \(p \leq 1\), as in this exercise, \(p = 0.75\) leads to divergence, indicating the series does not sum to a finite number.
• The p-value analysis is crucial in quickly identifying series behavior without needing to sum an infinite number of terms manually.

Using p-value analysis effectively allows for efficient evaluation of complex mathematical series, aiding in theoretical and applied mathematics.