Problem 22 - Thomas Calculus

Question

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{5^{n}}{4^{n}+3} $$

Step-by-Step Solution

Answer

The series diverges because the terms do not approach zero as \( n \to \infty \).

Step 1: Identify the Series

We are given the series \( \sum_{n=1}^{\infty} \frac{5^n}{4^n + 3} \). Our task is to determine if this series converges or diverges by examining its terms as \( n \) approaches infinity.

Step 2: Analyze the Terms

Look at the terms of the series: \( a_n = \frac{5^n}{4^n + 3} \). As \( n \) becomes very large, the term \( 4^n + 3 \) behaves similarly to \( 4^n \). Hence, \( a_n \approx \frac{5^n}{4^n} = \left(\frac{5}{4}\right)^n \).

Step 3: Apply the Ratio Test

The Ratio Test is helpful for series with terms like \( \left(\frac{5}{4}\right)^n \). For the series \( \sum a_n \), it states that if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \), the series converges if \( L < 1 \) and diverges if \( L > 1 \).

Step 4: Calculate the Ratio

Calculate \( \frac{a_{n+1}}{a_n} \):\[\frac{a_{n+1}}{a_n} = \frac{\frac{5^{n+1}}{4^{n+1} + 3}}{\frac{5^n}{4^n + 3}}\]This simplifies to \[\frac{5 \cdot 4^n + 3}{4^{n+1} + 3} \approx \frac{5}{4}\]as \( n \to \infty \).

Step 5: Determine Convergence or Divergence

As \( \lim_{n \to \infty} \frac{5 \cdot 4^n+3}{4^{n+1}+3} = \frac{5}{4} \), and \( \frac{5}{4} > 1 \). According to the Ratio Test, since \( L > 1 \), the series \( \sum_{n=1}^{\infty} \frac{5^n}{4^n + 3} \) diverges.

Key Concepts

Ratio Test

The Ratio Test is a handy tool for determining whether a series converges or diverges, especially when dealing with exponential expressions. It involves examining the limit of the ratio of successive terms in the series.
In mathematical terms, for a series \( \sum a_n \), the Ratio Test requires calculating \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \).

• If \( L < 1 \), the series converges.
• If \( L > 1 \), the series diverges.
• If \( L = 1 \), the test is inconclusive.

The Ratio Test is particularly effective when the fraction involves terms like \( \left(\frac{5}{4}\right)^n \) or other exponentials, because the ratio simplifies nicely. In our example, the series \( \sum_{n=1}^{\infty} \frac{5^n}{4^n + 3} \) shows a simplified form \( \left(\frac{5}{4}\right)^n \).
By calculating \( \lim_{n \to \infty} \left| \frac{5 \cdot 4^n + 3}{4^{n+1} + 3} \right| \), it's determined that the series diverges since \( L > 1 \). This simplifies understanding by reducing complex series to a singular condition.

Divergence

When discussing series in mathematics, a major point of interest is knowing if a series is converging to a particular value or diverging. Divergence means that as more terms are added, the series does not settle towards a specific number. Instead, it becomes unbounded or oscillates without reaching a fixed point.
To determine divergence, we compare the growth rate of the terms as \( n \) approaches infinity. If \( a_n \), or the terms of the series, grow larger without bound or fail to approach zero, the series is likely to diverge. In many cases, the Ratio Test is employed to identify divergence, especially for series with rapidly growing terms.
In our example, the emergent form \( \left(\frac{5}{4}\right)^n \) indicates divergence because the ratio \( \frac{5}{4} \) exceeds 1. This suggests exponential growth, meaning the series increases indefinitely rather than stabilizing, hence diverging.

Exponential Series

An exponential series is composed of terms that include exponential expressions, such as \( x^n \) where \( x \) is a base and \( n \) is the exponent. These series demonstrate remarkable properties and are central to many mathematical theories due to their growth patterns.
For instance, a series like \( \sum \left(\frac{5}{4}\right)^n \) showcases exponential growth because the base (\( \frac{5}{4} \)) is greater than 1, which implies the terms of the series expand exponentially.
Such series need specific methods, like the Ratio Test, to analyze their behavior. While some exponential series converge depending on their base and term structure, others like \( \left(\frac{5}{4}\right)^n \), can diverge if their base exceeds 1.
When analyzing series in exercises, spotting exponential terms helps in deciding the method to apply, predicting the series behavior, and applying the proper tests to confirm convergence or divergence.