Sum of an Infinite Geometric Series with First Term 100 and Common Ratio 12
Find the sum of an infinite geometric series where the first term is 100 and the common ratio is .
The sum of an infinite geometric series is a foundational topic in calculus and analysis, touching upon the concept of sequences and series. This type of problem focuses on the behavior of numbers in a sequence where each term after the first is derived by multiplying the previous term by a fixed number known as the common ratio. Determining the sum of an infinite series relies on the critical condition that this common ratio has an absolute value less than one. When this condition is satisfied, the series converges to a finite value. Understanding why and how this convergence occurs is key to tackling similar problems.
To solve problems like this, it's important to grasp the formula for the sum of an infinite geometric series: , where 'a' is the first term and 'r' is the common ratio. This formula is derived from the properties of geometric progressions and the limits of series. Recognizing when a series converges and applying this formula simplifies finding sums, even when the series continues indefinitely. Furthermore, this concept is widely applicable in various fields such as economics, physics, and computer science, illustrating the utility of mathematical theory in practical scenarios.
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