Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems , determine whether the series converges or diverges. Explain your reasoning. $$ \frac{3^{3}}{3}+\frac{4^{4}}{4}+\frac{5^{5}}{5}+\cdots $$

Step-by-Step Solution

Answer

The given series diverges.

Step 1: Identify the Series

The given series is \(\frac{3^{3}}{3}+\frac{4^{4}}{4}+\frac{5^{5}}{5}+\cdots \). It can be simplified and rewritten as \(\sum_{n=3}^{\infty} \frac{n^{n}}{n}\).

Step 2: Simplify the Series

The series simplifies further to \(\sum_{n=3}^{\infty} n^{n-1}\).

Step 3: Identify the p value

Since 'n-1' is part of the exponent in the series, the 'p' value in this case is equivalent to 'n-1'.

Step 4: Determine the Convergence or Divergence

The p-series statement of convergence defines that the series converges if \( p > 1 \). Since 'n-1' could be equal to or less than 1 (when n=2 or n=1), the resulting p-series could diverge.

Key Concepts

P-Series Test

The P-Series Test is a very useful tool when determining whether a particular series converges or diverges. It applies specifically to series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). In this kind of series, \( p \) represents a positive constant that influences the behavior of the series.The test is quite straightforward:

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

This rule helps simplify the complex task of determining the behavior of a series into understanding the value of \( p \). In the context of our problem, while the expression itself isn’t exactly a standard p-series, the idea of the exponent’s behavior helps identify the divergence of the series. Remember, the exponent needs to reflect \( p > 1 \) for convergence, which doesn't hold true for this scenario.

Exponential Series

An Exponential Series involves terms that are represented by exponential growth patterns. These series can take many forms, but they often demonstrate rapid growth due to the nature of exponentiation.In the problem provided, each term follows an exponential pattern, \( n^n \), divided by \( n \). This structure exemplifies how quickly terms grow in such a series.The concept to note here is that exponential growth can rapidly increase term values, leading generally to a divergent outcome unless other factors counterbalance this growth. The key takeaway is understanding how the base and exponent impact the term's magnitude, which ultimately influences convergence or divergence.

Convergent Series

Convergent Series are those whose terms approach a specific limit as \( n \) becomes very large. In simple terms, as you continue to add more and more terms, the sum approaches a finite value.A series converges if the sum of its terms stays bounded and doesn't increase infinitely. This characteristic is what makes convergence calculable and meaningful in many mathematical contexts. In practical terms, if you are looking at a sequence of numbers and the addition of subsequent numbers fails to alter the overall sum significantly once you reach a certain point, you're likely looking at a convergent series. Remember, convergent doesn't necessarily mean the terms themselves become zero; rather, their sum does not run to infinity.

Divergent Series

In contrast to convergent series, Divergent Series are those where the sum does not approach a finite limit. Instead, as you add more terms, the total sum continues to grow indefinitely.Divergence is essentially the failure of a series to achieve convergence. Several factors can cause a series to diverge:

• Exponential growth of terms.
• Failure of the sequence of partial sums to stabilize.

In the original exercise, the series growth was dominated by the exponential component \( n^{n-1} \), leading to terms that escalate quickly and cause divergence. This means that no matter how many terms you add, the series will not settle on a specific sum, highlighting the richness of divergent behavior in mathematical analysis.