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Understanding how to find the sum of an infinite geometric series is a fundamental concept in mathematics, particularly in calculus and algebra. This step-by-step guide will help you grasp the process clearly and confidently.
What Is an Infinite Geometric Series?
An infinite geometric series is a sum of infinitely many terms where each term is a constant multiple of the previous one. It has the form:
S = a + ar + ar2 + ar3 + …
Here, a is the first term, and r is the common ratio between terms.
Conditions for the Sum to Exist
The sum of an infinite geometric series converges (approaches a finite value) only if the absolute value of the common ratio is less than 1:
|r| < 1
Deriving the Sum Step-by-Step
Suppose the sum of the series is S. Then, we write:
S = a + ar + ar2 + ar3 + …
Multiply both sides of the equation by r:
rS = ar + ar2 + ar3 + ar4 + …
Subtract the second equation from the first:
S – rS = a
Factor out S on the left side:
S(1 – r) = a
Finally, solve for S:
S = \frac{a}{1 – r}
Conclusion
The formula S = \frac{a}{1 – r} gives the sum of an infinite geometric series when |r| < 1. Understanding this derivation helps you see why the sum converges and how the ratio influences the total.
Mastering this concept is essential for advanced studies in mathematics and related fields, providing a foundation for understanding limits, series, and calculus.