Question

State whether the given series converges or diverges. $$\sum_{n=0}^{\infty} \frac{1}{5^{n}}$$

Short answer

The series converges because it is a geometric series with \(|r| < 1\).

Step-by-step solution

Identify the Type of Series

The given series \( \sum_{n=0}^{\infty} \frac{1}{5^{n}} \) is a geometric series because each term is a constant multiple (common ratio) of the previous term. It has the form \( ar^n \) where \( a = 1 \) and \( r = \frac{1}{5} \).

Determine the Ratio

In a geometric series \( \sum_{n=0}^{\infty} ar^n \), the ratio \( r \) is crucial. For this series, \( r = \frac{1}{5} \).

Apply the Convergence Test for Geometric Series

A geometric series converges if the absolute value of the ratio \( |r| < 1 \). Here, \( |r| = \left| \frac{1}{5} \right| = \frac{1}{5} \), which is less than 1.

Conclusion About Convergence

Since \( |r| < 1 \), the series \( \sum_{n=0}^{\infty} \frac{1}{5^{n}} \) converges.

Key concepts

Series Convergence

In mathematics, a series is the sum of the terms of a sequence. Understanding whether a series converges or diverges is paramount. **Series convergence** refers to a series approaching a specific finite sum as the number of terms increases, and it’s determined by analyzing the series' nature. For instance, the given series \( \sum_{n=0}^{\infty} \frac{1}{5^{n}} \) particularly converges.

A series like this can converge if the terms being added decrease and approach zero over time. When studying this, we're often interested in what happens as 'n' goes to infinity. In the case of the geometric series presented, it's clear that because the multiplier or the common ratio (r) is less than 1, the series converges. Specifically, for a geometric series to converge, the absolute value of its common ratio must be less than 1, which in this case, \( \left| \frac{1}{5} \right| \), satisfies this condition. Therefore, the infinite sum of the terms involved will accumulate to a finite total.

Convergence Test

A **convergence test** is a method used to determine if a series converges or diverges. There are various tests available, but for geometric series, a specialized test exists. For a geometric series with a common ratio 'r', the test is simple:

• If \( |r| < 1 \), the series converges.
• If \( |r| \geq 1 \), the series diverges.

In our example, the convergence test is applied by noting the common ratio \( r = \frac{1}{5} \). Since \( |r| \) is clearly less than 1, the series indeed converges. This test provides a quick method for checking geometric series, making it easier to determine the behavior of an infinite series without having to sum it directly. Understanding the convergence test helps when dealing with series as it tells us about the potential behavior of series in terms of bounds and limits.

Infinite Series

An **infinite series** is a sum of infinitely many terms. It's typically expressed in the form \( \sum_{n=0}^{\infty} a_n \). The focus is on what happens to this sum as more and more terms are added. Infinite series can be daunting at first, but they represent a crucial part of mathematical analysis and many real-world applications.

A geometric series such as \( \sum_{n=0}^{\infty} \frac{1}{5^{n}} \) provides a perfect illustration of an infinite series. What makes this specific series interesting is its behavior as infinite terms are summed. Despite having an infinite number of terms, the finite convergence described earlier helps us understand its manageable behavior.

Infinite series are used for various practical applications including mathematical calculations, physics, and engineering solutions by approximating values too complex for straightforward computation. Realizing how these conceptual elements interconnect can aid students in seeing both the utility and beauty of mathematics.