Geometric Series (Sum infinite and finite geometric series and categorize geometric series)

Geometric Series

A geometric series is the sum of the terms in a geometric sequence, where each term after the first is found by multiplying the previous term by a constant called the common ratio. The general form of a geometric sequence is:

Where:

• $a$ is the first term.
• $r$ is the common ratio (the factor by which each term is multiplied to get the next term).
• $n$ is the number of terms.

Geometric Series:

A geometric series is the sum of the terms in a geometric sequence. If we sum the first $n$ terms of a geometric sequence, the result is a finite geometric series. The formula for the sum of the first $n$ terms of a finite geometric series is:

Where:

• $S_n$ is the sum of the first $n$ terms.
• $a$ is the first term.
• $r$ is the common ratio.
• $n$ is the number of terms.

Infinite Geometric Series:

An infinite geometric series is the sum of the terms in a geometric sequence that goes on indefinitely. The sum of an infinite geometric series is only finite (i.e., converges) when the absolute value of the common ratio $|r|$ is less than 1. In such cases, the sum of the infinite series is:

If $|r| \geq 1$, the series diverges and does not have a finite sum.

Key Categories of Geometric Series:

1. Finite Geometric Series: A finite geometric series is the sum of the first $n$ terms of a geometric sequence. The formula to find the sum of a finite geometric series is:

   $$S_n = \frac{a(1 - r^n)}{1 - r}, \quad \text{for} \quad r \neq 1$$

   Example: Find the sum of the first 4 terms of the geometric series with the first term $a = 3$ and the common ratio $r = 2$.

   The series is:

   $$3, 6, 12, 24, \ldots$$

   Using the formula for the sum of a finite geometric series:

   $$S_4 = \frac{3(1 - 2^4)}{1 - 2} = \frac{3(1 - 16)}{-1} = \frac{3(-15)}{-1} = 45$$

   So, the sum of the first 4 terms is 45.

2. Infinite Geometric Series: An infinite geometric series is the sum of an infinite number of terms. The sum converges if the absolute value of the common ratio $|r|$ is less than 1. The formula for the sum of an infinite geometric series is:

   $$S_{\infty} = \frac{a}{1 - r}, \quad \text{for} \quad |r| < 1$$

   Example: Find the sum of the infinite geometric series where $a = 5$ and $r = \frac{1}{2}$.

   The series is:

   $$5, \frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots$$

   Using the formula for the sum of an infinite geometric series:

   $$S_{\infty} = \frac{5}{1 - \frac{1}{2}} = \frac{5}{\frac{1}{2}} = 10$$

   So, the sum of the infinite geometric series is 10.

Geometric Series Convergence and Divergence:

1. Convergent Infinite Geometric Series: When $|r| < 1$, the infinite geometric series converges to a finite sum. This means that the sum of the series approaches a finite value as the number of terms increases.

2. Divergent Infinite Geometric Series: When $|r| \geq 1$, the infinite geometric series diverges. This means that the sum grows without bound as more terms are added.

   • If $r = 1$, each term in the series is equal to $a$, so the sum is just $a + a + a + \ldots$, which grows indefinitely.
   • If $|r| > 1$, the terms get larger in magnitude, and the sum grows without bound.

Examples:

Example 1: Finite Geometric Series

Find the sum of the first 6 terms of the geometric series with $a = 4$ and $r = \frac{1}{3}$.

The series is:

Using the formula for the sum of a finite geometric series:

So, the sum of the first 6 terms is approximately 2.

Example 2: Infinite Geometric Series

Find the sum of the infinite geometric series with $a = 7$ and $r = \frac{1}{4}$.

The series is:

Since $|r| = \frac{1}{4} < 1$, the series converges. Using the formula for the sum of an infinite geometric series:

So, the sum of the infinite geometric series is approximately $9.33$.

Summary:

• A geometric series is the sum of the terms of a geometric sequence.

• The sum of the first $n$ terms of a finite geometric series is given by:

  $$S_n = \frac{a(1 - r^n)}{1 - r}, \quad \text{for} \quad r \neq 1$$

• The sum of an infinite geometric series is given by:

  $$S_{\infty} = \frac{a}{1 - r}, \quad \text{for} \quad |r| < 1$$

• Convergence occurs when $|r| < 1$, and divergence occurs when $|r| \geq 1$.