Problem 18 - Thomas Calculus

Question

Which of the series in Exercises \(17-56\) converge, and which diverge? Use any method, and give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{3}{n+\sqrt{n}} $$

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{3}{n+\sqrt{n}} \) diverges.

Step 1: Analyze the Series Type

The given series is \( \sum_{n=1}^{\infty} \frac{3}{n+\sqrt{n}} \). This series is not a basic geometric series or a simple p-series, but it resembles a type of p-series once simplified due to the nature of the denominator.

Step 2: Simplify the Expression

Consider the expression \( \frac{3}{n+\sqrt{n}} \). We can rewrite the denominator as \( n(1+\frac{1}{\sqrt{n}}) \). For large \( n \), \( \sqrt{n} \) grows much slower than \( n \), so we approximate: \( \frac{3}{n+\sqrt{n}} \approx \frac{3}{n} \) for very large \( n \).

Step 3: Comparison with a Known Series

Use the Limit Comparison Test with the series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is a harmonic series known to diverge. Compute the limit: \( \lim_{n \to \infty} \frac{\frac{3}{n+\sqrt{n}}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{3n}{n+\sqrt{n}} \).

Step 4: Calculate the Limit

Simplify further: \( \lim_{n \to \infty} \frac{3n}{n+\sqrt{n}} = \lim_{n \to \infty} \frac{3n}{n(1+\frac{1}{\sqrt{n}})} = \lim_{n \to \infty} \frac{3}{1+\frac{1}{\sqrt{n}}} = 3 \).

Step 5: Apply the Limit Comparison Test

Since \( \lim_{n \to \infty} \frac{3n}{n+\sqrt{n}} = 3 \) and is finite and non-zero, it indicates that the series \( \sum_{n=1}^{\infty} \frac{3}{n+\sqrt{n}} \) behaves like \( \sum_{n=1}^{\infty} \frac{1}{n} \). As the harmonic series diverges, so does the original series.

Key Concepts

Harmonic Series

The harmonic series is foundational when studying series convergence. It is characterized by its formula: \[ \sum_{n=1}^{\infty} \frac{1}{n} \]This series is important because it is a classic example of a divergent series. Despite its simple form, as you keep adding terms, the series keeps growing larger without bound. For any finite sum of terms, the harmonic series diverges. This means it does not settle at a finite value, no matter how many terms you add.

• In practical terms, even though each term becomes smaller, the sum doesn't stabilize.
• This behavior is insightful for recognizing similar series.

Understanding this series helps identify other series that might also demonstrate divergence. In the exercise, we compared the given series to a harmonic series to show it also diverges.

Limit Comparison Test

The limit comparison test is a powerful tool for analyzing the convergence of series. It helps us compare an unfamiliar series with a more familiar one.Let's say we have two series: \[ a_n \] and \[ b_n \]. If \[ \lim_{n \to \infty} \frac{a_n}{b_n} = c \] where \( c \) is a positive and finite constant, both series either converge or diverge together.

• This method is most useful when dealing with series that are difficult to analyze directly.
• In our exercise, we used it with the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) to examine \( \sum_{n=1}^{\infty} \frac{3}{n+\sqrt{n}} \).

By computing \[ \lim_{n \to \infty} \frac{\frac{3}{n+\sqrt{n}}}{\frac{1}{n}} \]and finding it equal to 3, we determined that both series behave similarly, confirming divergence of the target series.

P-Series

The focus of the p-series provides further insight into series convergence. A p-series can be written as: \[ \sum_{n=1}^{\infty} \frac{1}{n^p} \] where the series converges if \( p > 1 \) and diverges if \( p \leq 1 \). Understanding this rule helps in evaluating series with terms that resemble those in a p-series.

• In analyzing our original series, we simplified it to resemble a p-series with \( p = 1 \).
• This simplification suggested behavior similar to the harmonic series, as they both share the same \( p \) value.

This robust framework not only aids in assessing series with explicit \( n^p \) terms but also others that can be transformed or approximated into a similar form. This method often grants clearer insights into convergence and divergence behaviors.