Problem 34 - Algebra 2

Question

Evaluate each infinite series that has a sum. $$ \sum_{n=1}^{\infty}\left(\frac{1}{5}\right)^{n-1} $$

Step-by-Step Solution

Answer

So, the sum of the given infinite series is 1.25.

Step 1: Identify 'a' and 'r'

First, identify the first term (a) and the common ratio (r) for the geometric series. Here, we can see that the first term (a) is 1, and the common ratio (r) is \(\frac{1}{5}\)

Step 2: Verify |r| < 1

To determine if the series converges and the sum can be found, check if the absolute value of the common ratio (r) is less than 1. In this case, since \(\left|\frac{1}{5}\right| < 1\), the series is convergent and the sum of the infinite series can be found using the formula.

Step 3: Calculate the sum

Now, plug 'a' and 'r' into the formula \(S = \frac{a}{1 - r}\) to get the sum of the series. Thus, the sum \(S = \frac{1}{1 - \frac{1}{5}} = \frac{1}{\frac{4}{5}} = \frac{5}{4} = 1.25\)

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (\(r\)). Geometric series can either be finite, where they have a limited number of terms, or infinite, stretching endlessly.
For example, the infinite geometric series in the exercise is represented as \(\sum_{n=1}^{\infty} \left(\frac{1}{5}\right)^{n-1}\), which means it starts from the term 1 and continues infinitely by multiplying each subsequent term by \(\frac{1}{5}\).
Identifying the first term \(a\) and the common ratio \(r\) is the first step in solving problems involving geometric series. In our case, \(a = 1\) and \(r = \frac{1}{5}\). By knowing these, we can understand the structure of the series and work towards evaluating its sum.

Convergence

Convergence in an infinite series occurs when the series approaches a finite value as the number of terms increases indefinitely. For a geometric series, convergence is heavily dependent on the value of the common ratio.
A geometric series converges only if the absolute value of the common ratio \(|r|\) is less than one. This ensures that each term gets smaller, allowing the sum to approach a specific limit instead of increasing indefinitely.

• In the exercise, we verify convergence by calculating \(|\frac{1}{5}| < 1\), which confirms the series converges.
• Thus, the series approaches a specific sum despite having an infinite number of terms.

Understanding convergence helps in determining whether a sum can be found and how we apply formulas to solve the series.

Sum of a Series

The sum of an infinite geometric series can be calculated using a special formula when the series converges. The formula is:
\[ S = \frac{a}{1 - r} \]
where \(S\) is the sum, \(a\) is the first term, and \(r\) is the common ratio.
This formula only applies to convergent geometric series, where \(|r| < 1\), as it relies on the concept of convergence to ensure a finite sum.
In the exercise example, using the calculated values \(a = 1\) and \(r = \frac{1}{5}\), we substitute these into the formula:

• Calculate the denominator: \,\(1 - \frac{1}{5} = \frac{4}{5}\)
• Plug values into the formula: \(S = \frac{1}{\frac{4}{5}} = \frac{5}{4} = 1.25\)

Through these steps, we've determined the sum of the series, which is 1.25. Understanding this calculation process is crucial for solving any infinite geometric series.