Problem 38 - Algebra and Trigonometry

Question

Find the sum of each infinite geometric series. $$ 1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots $$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is \( \frac{4}{3} \)

Step 1: Identify the First Term and Common Ratio

From the given series, the first term \(a\) is 1 and the common ratio \(r\) is \( \frac{1}{4} \)

Step 2: Plug Values into the Formula

Substitute the identified values into the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - r} \). This gives us \( S = \frac{1}{1 - \frac{1}{4}} \)

Step 3: Simplify the Equation

Simplify the equation to find the sum of the series. Simplify the denominator first: \( 1 - \frac{1}{4} = \frac{3}{4} \). The equation becomes \( S = \frac{1}{\frac{3}{4}} \). By dividing 1 by \( \frac{3}{4} \) we get \( S = \frac{4}{3} \)

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Geometric series can be infinite or finite. In our example, the series is infinite:

• 1
• \( \frac{1}{4} \)
• \( \frac{1}{16} \)
• \( \frac{1}{64} \)

... and so on as it continues to infinity. A geometric series has a special property where you can easily calculate the sum if it involves an infinite number of terms. This property comes from the series approaching a certain limit as you continue adding more terms, provided the common ratio is between -1 and 1.

Sum Formula

The sum formula for an infinite geometric series is used to find the sum of all the terms in the series when it is infinite. You can only use this formula when the common ratio's absolute value is less than one (\( |r| < 1 \)). This is because, with a common ratio of less than one, the series converges to a finite limit. The sum of an infinite geometric series can be found using the formula:\[S = \frac{a}{1 - r}\]where:- \(S\) is the sum of the series.- \(a\) is the first term of the series.- \(r\) is the common ratio.

Common Ratio

The common ratio in a geometric series is the factor you multiply each term by to get the next term. It's a crucial part of understanding and calculating the series. In the series provided:

• The first term is \(1\).
• The next term is \(\frac{1}{4}\), found by multiplying the first term by the common ratio \( \frac{1}{4} \).

The common ratio is constant throughout the series. It determines how quickly or slowly the numbers in the series approach zero. For infinite geometric series, having a common ratio with an absolute value less than 1 ensures the series will have a finite sum.

First Term

The first term in a geometric series is the starting point of the series. It is denoted by \(a\) in formulas. Understanding the first term is the foundation for exploring further terms in the series.In our series:

• The first term \(a\) is \(1\).
• This first term is essential as it helps in calculating the series' sum when combined with the common ratio.

Once you have the first term and the common ratio, you are well equipped to use the sum formula for calculating the sum of an infinite geometric series, giving a complete picture of the series' overall behavior.