Problem 75 - Essential Calculus

Question

Use the divergence test given in Exercise 71 to show that the series diverges. $$ \sum_{n=1}^{\infty} \frac{n^{3}}{5 n^{4}+3} $$

Step-by-Step Solution

Answer

Using the divergence test, we can't conclusively say whether the series \(\sum_{n=1}^{\infty} \frac{n^{3}}{5 n^{4}+3}\) converges or diverges.

Step 1: Identify the sequence of terms

The sequence of terms of the given series is \(\frac{n^{3}}{5 n^{4}+3}\).

Step 2: Compute the limit as n approaches infinity

Calculate the limit of the sequence of terms as n approaches infinity using L’Hopital’s rule since it is in the form of \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). The rule states that: \[ \lim_{n \rightarrow \infty} \frac{f(n)}{g(n)} = \lim_{n \rightarrow \infty} \frac{f'(n)}{g'(n)} \] where \(f(n)\) and \(g(n)\) are differentiable on an open interval that contains \(n\), except possibly at \(n\) itself. Thus, calculate the derivatives: \(f'(n) = 3n^{2}\) and \(g'(n) = 20n^{3}\)

Step 3: Apply L’Hopital’s rule

Applying L'Hopital’s rule, we get \[ \lim_{n \rightarrow \infty} \frac{3n^{2}}{20n^{3}} = \lim_{n \rightarrow \infty} \frac{3}{20n} \]

Step 4: Evaluate the limit

As \(n\) approaches infinity, the limit of \(\frac{3}{20n}\) equals 0. Thus, the limit of the sequence of terms of the series is 0.

Step 5: Interpreting the result

As the limit of the sequence of terms equals zero, the divergence test doesn't conclude whether the series converges or diverges. Therefore, the series might converge or diverge, more investigation is needed.

Key Concepts

Infinite Series

An infinite series is a sum of an infinite sequence of numbers. In mathematical notation, it's often expressed as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the general term of the sequence. When we deal with infinite series, our goal is often to determine whether the series converges or diverges.
Convergence means that as you add more and more terms, the sum approaches a finite number. Divergence, on the other hand, means the sum does not settle at a finite number but instead grows without bound or oscillates indefinitely.
For example, the series \( \sum_{n=1}^{\infty} \frac{n^3}{5n^4+3} \) is an infinite series because it proposes adding up an infinite number of terms formed by the formula \( \frac{n^3}{5n^4+3} \). Determining whether this series converges or diverges is crucial in understanding its behavior. Not all series are easy to evaluate, which is why we have tests like the divergence test.

L'Hôpital's Rule

L'Hôpital's Rule is a method in calculus used to find limits of indeterminate forms. Indeterminate forms commonly appear as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). In such cases, directly substituting into the limit results in an undefined expression.
To apply L'Hôpital's Rule, differentiate the numerator and the denominator of the function separately and then take the limit again. This can simplify the expression, allowing for evaluation. Mathematically, it is expressed as:

• \( \lim_{n \rightarrow \infty} \frac{f(n)}{g(n)} = \lim_{n \rightarrow \infty} \frac{f'(n)}{g'(n)} \)

This rule was applied to \( \frac{n^3}{5n^4+3} \) in our problem. By differentiating separately to get \( f'(n) = 3n^2 \) and \( g'(n) = 20n^3 \), we were able to simplify the limit to \( \lim_{n \rightarrow \infty} \frac{3}{20n} \), which is zero as \( n \) approaches infinity. However, reaching a limit of zero does not automatically indicate whether the series converges.

Convergence and Divergence Tests

Convergence and divergence tests are tools used to determine if an infinite series converges or diverges. In this context, the divergence test is crucial, specifically for providing initial insights into the behavior of a series.
The divergence test, sometimes called the nth-term test, simply states: if the limit of the sequence of terms (\( a_n \)) is not zero, then the series \( \sum a_n \) definitely diverges. However, a limit of zero does not confirm convergence; it merely indicates that the test is inconclusive on its own.

• Convergence Test: Need additional tests if divergence test does not confirm divergence.
• Divergence Test: If \( \lim_{n \rightarrow \infty} a_n eq 0 \), the series diverges.

This is exactly what we see in the series \( \sum_{n=1}^{\infty} \frac{n^3}{5n^4+3} \) — the divergence test results in zero, which means other tests like the Ratio Test, Root Test, or Integral Test might be applied to further determine convergence.