Question

State whether the given series converges or diverges. $$\sum_{n=0}^{\infty} \frac{6^{n}}{5^{n}}$$

Short answer

The series diverges because the common ratio is greater than 1.

Step-by-step solution

Recognize the Series Type

The series given is \( \sum_{n=0}^{\infty} \frac{6^n}{5^n} \). Notice that each term of the series can be rewritten as \( \left( \frac{6}{5} \right)^n \). This is a geometric series, which has the form \( \sum_{n=0}^{\infty} ar^n \) with \( a \) being the first term and \( r \) the common ratio.

Identify the First Term and Common Ratio

For the series \( \sum_{n=0}^{\infty} \left( \frac{6}{5} \right)^n \), identify that the first term \((a)\) is \( \left( \frac{6}{5} \right)^0 = 1 \). The common ratio \( r \) is \( \frac{6}{5} \).

Apply the Convergence Condition for Geometric Series

A geometric series converges if the absolute value of the common ratio \( |r| < 1 \). If \( |r| \geq 1 \), the series diverges. Here, \( r = \frac{6}{5} \) and thus \( |r| = \frac{6}{5} > 1 \).

Conclusion Based on Convergence Test

Since \( |r| = \frac{6}{5} \) is greater than 1, the given series diverges by the geometric series test.

Key concepts

series convergence

In mathematics, series convergence is a fundamental concept that tells us if the sum of an infinite sequence of terms will reach a finite value. Convergence in a series means that as you add up more and more terms of the series, the total sum approaches a specific number. This number is the limit of the series.

For example, with the geometric series under investigation, we look to see if the sum \(\sum_{n=0}^{\infty} \left( \frac{6}{5} \right)^n \) grows indefinitely or settles down to a particular number. This involves understanding the terms and recognizing a pattern or rule. If such a limit does not exist, this is called divergence, meaning the sum doesn’t settle at any value.

In simpler scenarios, such as series with beautiful fractional patterns, the decision whether they converge or not becomes easier once you establish the main properties—like the common ratio.

common ratio

A geometric series is defined by each term being a constant multiple of the previous term. This constant multiple is known as the "common ratio." In our example, the geometric series is represented by \(\sum_{n=0}^{\infty} \left( \frac{6}{5} \right)^n \).

The common ratio \(r\) is found by dividing any term in the series by the previous term, e.g., \(\frac{6}{5}^1 / \frac{6}{5}^0 = \frac{6}{5}\). This ratio remains constant throughout the series and determines much of the series' behavior.

Here, it is essential to highlight that the common ratio \(r\) affects the convergence of a series greatly. If \(|r| < 1\), the terms keep decreasing, pushing the series toward convergence. If \(|r| \geq 1\), the terms either grow out of bounds or move erratically, leading the series to diverge.

geometric convergence criteria

When dealing with geometric series, the convergence primarily depends on the common ratio \(r\). The geometric convergence criteria provide a straightforward rule:

• Converges if \(|r| < 1\)
• Diverges if \(|r| \geq 1\)

This criterion helps determine whether the sum of all terms in the series will approach a specific number (convergent), or keep growing indefinitely or oscillating (divergent).

In the solved exercise, the series \(\sum_{n=0}^{\infty} \left( \frac{6}{5} \right)^n \) had a common ratio of \(\frac{6}{5}\), which is greater than 1. According to the geometric convergence criteria, since \(|r| = \frac{6}{5} > 1\), the series diverges. This showcases a clear application of the criteria allowing us to classify the behavior of the series efficiently.