Geometric Series: Infinite Geometric Series Explained

♾️ Infinite Geometric Series: A Deep Dive

An infinite geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. Understanding when these series converge (approach a finite value) or diverge (grow without bound) is crucial. Let's explore this concept in detail.

🤔 Convergence vs. Divergence

The behavior of an infinite geometric series depends on its common ratio, denoted as 'r'.

• Convergence: If the absolute value of the common ratio is less than 1 (i.e., |r| < 1), the series converges. This means the sum approaches a finite value.
• Divergence: If the absolute value of the common ratio is greater than or equal to 1 (i.e., |r| ≥ 1), the series diverges. This means the sum does not approach a finite value and grows indefinitely.

📐 Formula for the Sum of a Convergent Infinite Geometric Series

When an infinite geometric series converges, we can calculate its sum using the following formula:

$$S = \frac{a}{1 - r}$$

Where:

• $S$ is the sum of the infinite series.
• $a$ is the first term of the series.
• $r$ is the common ratio (and |r| < 1 for convergence).

✍️ Examples

Example 1: Convergent Series

Consider the series: 1 + 1/2 + 1/4 + 1/8 + ...

Here, $a = 1$ and $r = 1/2$. Since |1/2| < 1, the series converges. Let's find its sum:

$$S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2$$

Thus, the sum of the series is 2.

Example 2: Divergent Series

Consider the series: 1 + 2 + 4 + 8 + ...

Here, $a = 1$ and $r = 2$. Since |2| ≥ 1, the series diverges. It does not have a finite sum.

Example 3: Another Convergent Series

Consider the series: 5 - 5/3 + 5/9 - 5/27 + ...

Here, $a = 5$ and $r = -1/3$. Since |-1/3| < 1, the series converges. Let's find its sum:

$$S = \frac{5}{1 - (-\frac{1}{3})} = \frac{5}{\frac{4}{3}} = \frac{15}{4}$$

Thus, the sum of the series is 15/4 or 3.75.

💻 Code Example (Python)

Here's a Python code snippet to calculate the sum of a convergent geometric series:

def geometric_sum(a, r, n_terms):
    if abs(r) >= 1:
        return "Series diverges"
    else:
        return a / (1 - r)

# Example usage:
first_term = 1
common_ratio = 0.5

sum_of_series = geometric_sum(first_term, common_ratio, 100) #using 100 terms to approximate infinite series
print(f"The sum of the infinite geometric series is: {sum_of_series}")

💡 Key Takeaways

• Infinite geometric series converge only when the absolute value of the common ratio is less than 1 (|r| < 1).
• The sum of a convergent infinite geometric series can be found using the formula $S = \frac{a}{1 - r}$.
• Divergent geometric series do not have a finite sum.