Problem 29 - Algebra 2

Question

Evaluate each infinite geometric series. $$ 1-\frac{1}{5}+\frac{1}{25}-\frac{1}{125}+\ldots $$

Step-by-Step Solution

Answer

The sum of the infinite geometric series \(1 - 1/5 + 1/25 - 1/125 + \ldots\) is \(5/6\).

Step 1: Identify the first term

In the given series, take note of the first term: \(a = 1\)

Step 2: Identify the common ratio

The common ratio r can be identified by dividing the second term by the first, or the third term by the second, and so on, as the ratio is constant for a geometric series. In this case, \(r = -1/5\).

Step 3: Apply the formula for the sum of an infinite geometric series

Use the formula \(S = \frac{a}{1-r}\) and substitute the identified values. This gives: \(S = \frac{1}{1 - (-1/5)} = \frac{1}{1 + 1/5} = \frac{1}{6/5} = \frac{5}{6}\), after simplification.

Key Concepts

Sum of Series

In an infinite geometric series, the sum refers to the total of all terms in the series extending infinitely. These series have a unique property when the absolute value of the common ratio is less than one. This allows us to calculate what seems like an endless sum, resulting in a finite number. Even though the series goes on forever, its sum can ultimately be calculated. Think of it like approaching a destination that you never quite reach, yet you come infinitely close.

The series, in this example, is:

• 1 - \(\frac{1}{5}\) + \(\frac{1}{25}\) - \(\frac{1}{125}\) + \dots\

To find the sum of such a series, one must use a specific formula which involves the first term and the common ratio. This formula allows for the calculation of the series' sum, giving us a real number even though the series is infinite.

Common Ratio

The common ratio is a crucial component in determining the nature of a geometric series. It is the consistent factor that each term is multiplied by to get the next term. In an infinite geometric series, recognizing the common ratio is key to finding the sum.

To find the common ratio, simply divide any term in the series by the previous term. The ratio stays the same throughout.
For example,

• from the series 1, -\(\frac{1}{5}\), \(\frac{1}{25}\), -\(\frac{1}{125}\), the common ratio \(r\) is \(-\frac{1}{5}\). This is obtained by dividing \(-\frac{1}{5}\) by 1, \(\frac{1}{25}\) by \(-\frac{1}{5}\), and so on.

Having a common ratio where the absolute value is less than 1 ensures the series approaches a finite sum, despite having an infinite number of terms.

Geometric Series Formula

The geometric series formula is a powerful tool that allows you to calculate the sum of an infinite series. The sum of an infinite geometric series can be determined using this specific formula: \[ S = \frac{a}{1 - r} \]Here, \(S\) represents the sum of the series, \(a\) is the first term, and \(r\) is the common ratio. For this formula to apply, the absolute value of \(r\) must be less than one.

Let's apply it to our series:

• First term \(a\) is 1.
• Common ratio \(r\) is \(-\frac{1}{5}\).

Substitute these values into the formula:

\[ S = \frac{1}{1 - (-\frac{1}{5})} = \frac{1}{1 + \frac{1}{5}} = \frac{5}{6} \]The result, \(\frac{5}{6}\), is the sum of this infinite series. This reveals that despite its infinite nature, the series converges to a specific number.