Sum to Infinite Terms of GP Formula: Definition, Examples

Sum to Infinite Terms of GP Formula: We will examine the process of calculating the sum of an infinite Geometric Progression (GP). Contrary to the misconception that the sum of an infinite number of terms results in infinity, the sum of an infinite Geometric Progression (GP) is actually a finite number when the absolute value of the common ratio is less than 1. In the following discussion, we will explore the formula for calculating the sum of an infinite GP and provide a proof for this concept.

Sum of Infinite GP: What is it?

The sum of an infinite Geometric Progression (GP) can be determined by finding the sum of its first 'n' terms using the formula Sn = a(1 - r^n) / (1 - r), where 'a' is the first term, and 'r' is the common ratio. But what if we want to find the sum of all the terms of an infinite GP?

Consider the series: S = 1 + 1/2 + 1/4 + 1/8 +... with an infinite number of terms. Is this sum finite or infinite? This is a GP with a common ratio of magnitude less than 1, and it's important to note that all such GPs have finite sums. Let's explore this specific case and find the sum.

We start with the original series for S:

S = 1 + 1/2 + 1/4 + 1/8 +... ___ (1)

Now, we multiply each term on both sides by the common ratio (1/2):

S/2 = 1/2 + 1/4 + 1/8 +... ___ (2)

Next, we subtract the series (2) from series (1):

S - (S/2) = 1

Now, solve for S:

S/2 = 1

S = 2

Hence, the sum of the infinite terms in this GP is 2.

You might wonder why the sum of an infinite GP is a finite number in this case. It's because when the common ratio is less than 1, the terms in the series become progressively smaller. To illustrate this, let's calculate the sum of the first few terms:

S3 = 1 + 1/2 + 1/4 = 1.75

S4 = 1 + 1/2 + 1/4 + 1/8 = 1.875

S5 = 1 + 1/2 + 1/4 + 1/8 + 1/16 = 1.9375

As we add more terms, the sum approaches but never exceeds 2. This is because the subsequent terms, like 1/1024, 1/2048, 1/4096, and so on, are extremely small, almost approaching zero when expressed as decimals. Consequently, the sum of the infinite terms of the GP converges to a finite number, which in this case is 2.

Demonstrating the Convergence

To illustrate the concept further, we can calculate the sum of the first few terms:

S3 = 1 + 1/2 + 1/4 = 1.75

S4 = 1 + 1/2 + 1/4 + 1/8 = 1.875

S5 = 1 + 1/2 + 1/4 + 1/8 + 1/16 = 1.9375

...and so on.

These sums, as observed, consistently approach 2, showcasing the convergence of the infinite GP. This convergence occurs because as we progress, the terms in the GP become extremely small, resembling values close to zero.

Sum of Infinite GP for |r| < 1 and |r| ≥ 1

When |r| < 1: A Formula for Infinite GP

To calculate the sum of an infinite Geometric Progression (GP) when |r| < 1, we can derive a formula. Let's consider a GP with its first term 'a' and a common ratio 'r' where |r| < 1. The sum of its infinite terms can be expressed as:

S = a + ar + ar² + ar³ + ... (1)

Now, let's multiply both sides by 'r':

rS = ar + ar² + ar³ + ... (2)

By subtracting equation (2) from equation (1), we get:

S - rS = a

Now, we can solve for 'S' by factoring out 'S' on the left side:

S(1 - r) = a

Finally, we can find 'S' by dividing both sides by (1 - r):

S = a / (1 - r)

This formula allows us to calculate the sum of an infinite GP when the common ratio 'r' is less than 1.

When |r| ≥ 1: Divergence of the Series

In contrast, when |r| ≥ 1, the behavior of an infinite GP changes. As illustrated in the example in the first section, when |r| < 1, the GP converges to a finite sum because the terms become infinitesimally small after a certain number of terms.

However, when |r| ≥ 1, the GP takes a different form, such as 2, 4, 6, 8, 16, 32, 64, 128, 256, 512, 1024, and so on. In this scenario, the terms increase rapidly and become very large numbers after a certain point in the series. Consequently, the sum of a GP with |r| ≥ 1 cannot be found, and the series diverges, meaning it doesn't converge to a finite value.

Infinite GP Sum Formulas

When |r| is less than 1 (|r| < 1), the infinite GP sum (S∞) can be expressed as a / (1 - r), with 'a' denoting the initial term and 'r' as the common ratio.

For |r| greater than or equal to 1 (|r| ≥ 1), the infinite GP sum diverges and is represented as S∞ = ±∞.

Sum to Infinite Terms of GP Formula Keynote

To summarize, the sum of infinite GP formulas can be represented as follows:

When |r| < 1: S∞ = a / (1 - r)

When |r| ≥ 1: S∞ = ±∞

These formulas allow us to determine whether the sum of an infinite GP is finite or infinite based on the value of the common ratio 'r.'