Problem 24 - Thomas Calculus

Question

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{(-2)^{n+1}}{n+5^{n}} $$

Step-by-Step Solution

Answer

The series converges absolutely.

Step 1: Identify the Series

Consider the given series \( \sum_{n=1}^{\infty} \frac{(-2)^{n+1}}{n+5^{n}} \). This is an alternating series because the term \((-2)^{n+1}\) changes sign with each consecutive \(n\).

Step 2: Examine Absolute Convergence

To determine absolute convergence, we consider the series of the absolute values of its terms: \( \sum_{n=1}^{\infty} \frac{|(-2)^{n+1}|}{n+5^{n}} = \sum_{n=1}^{\infty} \frac{2^{n+1}}{n+5^{n}} \).

Step 3: Use the Ratio Test for Absolute Series

Apply the ratio test to \(\sum_{n=1}^{\infty} \frac{2^{n+1}}{n+5^{n}}\).Calculate the ratio: \( L = \lim_{n \to \infty} \left| \frac{2^{n+2}}{(n+1)5^{n+1}} \times \frac{n5^{n}}{2^{n+1}} \right| \).Simplify:\( L = \lim_{n \to \infty} \frac{2 \cdot n}{5(n+1)} = \frac{2}{5} \).Since \( \frac{2}{5} < 1 \), the series converges absolutely.

Step 4: Conclusion Based on Tests

Since the series of absolute values \( \sum_{n=1}^{\infty} \frac{2^{n+1}}{n+5^{n}} \) converges, the original series \( \sum_{n=1}^{\infty} \frac{(-2)^{n+1}}{n+5^{n}} \) converges absolutely.

Key Concepts

Alternating Series

An alternating series is one where the terms alternate in sign. For example, a series might have terms that are positive, negative, positive again, and so on, creating a sequence of alternating signs. Consider the given series:

• The general term is \( \frac{(-2)^{n+1}}{n+5^{n}} \), and you will notice that the term
• \((-2)^{n+1}\) changes sign as \(n\) increases.

This characteristic confirms it is an alternating series. Alternating series often converge even if the series of their absolute values do not.
The Alternating Series Test can be used to determine if an alternating series converges by checking two conditions:

• The absolute value of the terms decreases steadily.
• The limit of the absolute values of the terms as \( n \to \infty \) is zero.

When both conditions are met, the alternating series converges.

Absolute Convergence

A sequence is said to converge absolutely if the series formed by taking the absolute values of the terms converges.
In other words, we look at the series \( \sum \frac{|(-2)^{n+1}|}{n+5^n} = \sum \frac{2^{n+1}}{n+5^n} \).

• Absolute convergence implies that the series converges regardless of the sign changes, providing a stronger form of convergence.
• If a series converges absolutely, it must also converge normally.

Absolute convergence is a helpful property because it guarantees convergence without relying on the altering signs of the alternating series.

Ratio Test

The ratio test is a tool used to determine the absolute convergence of a series. To apply it to a series \( \sum a_n \), you compute the limit: \[ L = \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \]The series converges absolutely if \( L < 1 \), diverges if \( L > 1 \) or is inconclusive if \( L = 1 \).
For our series:

• We use the absolute series \( \sum \frac{2^{n+1}}{n+5^n} \).
• After computing, \( L = \frac{2}{5} \), which is less than 1.

So, the series converges absolutely. The ratio test becomes a reliable method to quickly assess absolute convergence, making it a popular test among students studying series.