The Geometric Sequence Infinite Sum
The sum of an infinite geometric sequence, also known as the geometric series, can be calculated if the common ratio (r) of the sequence satisfies the condition -1 < r < 1
The sum of an infinite geometric sequence, also known as the geometric series, can be calculated if the common ratio (r) of the sequence satisfies the condition -1 < r < 1. In this case, the sequence converges and has a finite sum. The formula to find the sum of an infinite geometric series is given by: S = a / (1 - r) where: S represents the sum of the series, a is the first term in the sequence, and r is the common ratio. To understand why this formula works, let's break it down step by step using the geometric sequence: a, ar, ar^2, ar^3, ar^4, ... To find the sum of this infinite series, we start by adding up the first n terms: S_n = a + ar + ar^2 + ar^3 + ... + ar^(n-1) Next, we multiply both sides of this equation by the common ratio (r): rS_n = ar + ar^2 + ar^3 + ... + ar^(n-1) + ar^n Now, if we subtract the second equation from the first one: S_n - rS_n = a - ar^n Notice that all the terms between ar and ar^n on the right side of the equation canceled out. Simplifying the equation, we have: S_n(1 - r) = a - ar^n Dividing both sides by (1 - r), we get: S_n = (a - ar^n) / (1 - r) Now, as n approaches infinity, the value of r^n becomes smaller and smaller (since -1 < r < 1), and ar^n approaches zero. Therefore, we have: S = a / (1 - r) This formula gives us the sum of an infinite geometric sequence as long as the common ratio (r) is between -1 and 1. If the common ratio is greater than or equal to 1 or less than -1, the series diverges, meaning it does not have a finite sum.
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