Formula For The Sum Of An Infinite Geometric Series

An infinite geometric series may seem like an abstract concept, but its formula holds the key to understanding convergence, limits, and various applications in mathematics and physics.

Understanding Geometric Series

A geometric series is a series where each term is multiplied by a constant value, known as the common ratio (r), to obtain the next term. For example, the sequence 2, 4, 8, 16, ... is a geometric sequence with a common ratio of 2. If we express this as a sum, we get the geometric series 2 + 4 + 8 + 16 + ....

In general terms, a geometric series can be written as:

a + ar + ar^2 + ar^3 + ...

where:

• a is the first term, and
• r is the common ratio.

An infinite geometric series is a geometric series that continues infinitely. For example:

1 + 1/2 + 1/4 + 1/8 + ...

Now, the interesting question arises: can we find the sum of such an infinite series? In some cases, the answer is yes, provided that the series converges.

Convergence vs. Divergence

Before diving into the formula, it is crucial to understand the concepts of convergence and divergence.

• Convergence: An infinite series converges if the sum of its terms approaches a finite value as the number of terms approaches infinity. In other words, as we add more and more terms, the sum gets closer and closer to a specific number.
• Divergence: An infinite series diverges if the sum of its terms does not approach a finite value. This means the sum either increases (or decreases) without bound, or it oscillates without settling on a specific number.

For a geometric series, the convergence or divergence depends entirely on the value of the common ratio, r:

• If |r| < 1 (i.e., -1 < r < 1), the series converges.
• If |r| ≥ 1, the series diverges.

Intuitively, when the absolute value of r is less than 1, each subsequent term becomes smaller and smaller, approaching zero. This allows the series to settle on a finite sum. Conversely, when the absolute value of r is greater than or equal to 1, the terms either stay the same size or increase, preventing the series from converging.

The Formula for the Sum of an Infinite Geometric Series

The sum (S) of an infinite geometric series, when it converges (|r| < 1), is given by the formula:

S = a / (1 - r)

where:

• S is the sum of the infinite geometric series.
• a is the first term.
• r is the common ratio.

Derivation of the Formula

To understand where this formula comes from, let's consider the sum of the first n terms of a geometric series, denoted as S_n:

S_n = a + ar + ar^2 + ar^3 + ... + ar^(n-1)

Now, multiply both sides of the equation by r:

rS_n = ar + ar^2 + ar^3 + ... + ar^(n-1) + ar^n

Subtract the second equation from the first:

S_n - rS_n = a - ar^n

Factor out S_n on the left side:

S_n(1 - r) = a - ar^n

Now, solve for S_n:

S_n = (a - ar^n) / (1 - r)

This is the formula for the sum of the first n terms. To find the sum of an infinite geometric series, we need to consider what happens as n approaches infinity.

If |r| < 1, then as n approaches infinity, r^n approaches zero. Therefore:

lim (n→∞) S_n = lim (n→∞) (a - ar^n) / (1 - r) = a / (1 - r)

Thus, the sum of the infinite geometric series is:

S = a / (1 - r)

Applying the Formula: Examples

Let's work through a few examples to illustrate how to use the formula:

Example 1:

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

• a = 1 (the first term)
• r = 1/2 (the common ratio)

Since |1/2| < 1, the series converges. Using the formula:

S = 1 / (1 - 1/2) = 1 / (1/2) = 2

Therefore, the sum of the infinite series is 2.

Example 2:

Consider the series: 3 - 6/5 + 12/25 - 24/125 + ...

• a = 3 (the first term)
• r = -2/5 (the common ratio)

Since |-2/5| < 1, the series converges. Using the formula:

S = 3 / (1 - (-2/5)) = 3 / (1 + 2/5) = 3 / (7/5) = 15/7

Therefore, the sum of the infinite series is 15/7.

Example 3:

Consider the series: 4 + 8 + 16 + 32 + ...

• a = 4 (the first term)
• r = 2 (the common ratio)

Since |2| ≥ 1, the series diverges and does not have a finite sum.

Example 4:

Find the sum of the infinite geometric series where a = 5 and r = 0.8.

Since |0.8| < 1, the series converges.

S = 5 / (1 - 0.8) = 5 / 0.2 = 25

Therefore, the sum of the infinite series is 25.

Practical Applications

The formula for the sum of an infinite geometric series is not just a theoretical concept; it has numerous practical applications in various fields:

• Economics: It is used to model the multiplier effect in economics, where an initial injection of spending can lead to a larger overall increase in economic activity.
• Physics: It appears in the calculation of probabilities in quantum mechanics and in the analysis of damped oscillations.
• Computer Science: It is used in analyzing the performance of algorithms and in data compression techniques.
• Mathematics: It is fundamental in calculus, complex analysis, and number theory.
• Finance: It can be used to calculate the present value of a perpetuity (a stream of payments that continues indefinitely).
• Repeating Decimals: Converting repeating decimals to fractions.

Example: Present Value of a Perpetuity

Suppose you are promised a payment of $1000 per year forever, and the interest rate is 5% per year. What is the present value of this perpetuity?

This can be modeled as an infinite geometric series:

PV = 1000 + 1000/(1.05) + 1000/(1.05)^2 + 1000/(1.05)^3 + ...

Here, a = 1000 and r = 1/1.05. Since |1/1.05| < 1, the series converges.

Using the formula:

PV = 1000 / (1 - 1/1.05) = 1000 / (0.05/1.05) = 1000 * (1.05/0.05) = 21000

Therefore, the present value of the perpetuity is $21,000.

Example: Converting Repeating Decimals to Fractions

Consider the repeating decimal 0.3333...

We can express this as an infinite geometric series:

0. 3 + 0.03 + 0.003 + 0.0003 + ...

Here, a = 0.3 and r = 0.1. Since |0.1| < 1, the series converges.

Using the formula:

S = 0.3 / (1 - 0.1) = 0.3 / 0.9 = 1/3

Therefore, 0.3333... is equivalent to the fraction 1/3.

Common Mistakes to Avoid

When working with the formula for the sum of an infinite geometric series, it's essential to be aware of potential pitfalls:

1. Forgetting to Check for Convergence: Always verify that |r| < 1 before applying the formula. If the series diverges, the formula will give an incorrect result.
2. Incorrectly Identifying a and r: Ensure you correctly identify the first term (a) and the common ratio (r). A mistake in either value will lead to an incorrect sum.
3. Misunderstanding the Formula: The formula S = a / (1 - r) only applies to infinite geometric series. Do not use it for finite series.
4. Sign Errors: Pay close attention to the signs of a and r. A negative sign in either value can significantly affect the result.
5. Assuming All Infinite Series Converge: Not all infinite series converge. The geometric series is a special case where convergence can be easily determined.

Advanced Concepts and Extensions

While the formula S = a / (1 - r) is straightforward, it's a gateway to more advanced concepts:

• Power Series: The concept of an infinite geometric series is closely related to power series, which are infinite series of the form Σ c_n(x - a)^n. Power series are used to represent functions and solve differential equations.
• Taylor and Maclaurin Series: These are special types of power series that provide a way to approximate functions using polynomials.
• Radius of Convergence: For power series, the radius of convergence determines the range of values for which the series converges.
• Zeno's Paradoxes: The concept of an infinite geometric series helps resolve some of Zeno's paradoxes, such as the paradox of Achilles and the tortoise, which involves an infinite series of decreasing distances.

Conclusion

The formula for the sum of an infinite geometric series is a powerful tool with wide-ranging applications. By understanding the concepts of convergence, divergence, and the derivation of the formula, you can effectively solve problems in various fields, from economics to physics. Remember to always check for convergence before applying the formula and be mindful of potential mistakes. As you delve deeper into mathematics, you'll find that this formula is a building block for more advanced concepts and techniques.