Infinite geometric series involve the summation of terms in a sequence multiplied by a common ratio. This text delves into the formula for calculating such series, distinguishing between convergent and divergent series based on the common ratio's absolute value. It also explores practical applications in various fields, demonstrating the significance of these series in both theoretical and practical contexts.
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An infinite geometric series is a summation of an endless sequence of terms, where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio
Common Ratio
The common ratio is a fixed, non-zero number that represents the multiplicative relationship between successive terms in a geometric sequence
Consistency
A geometric sequence is a structured list of numbers where the multiplicative relationship between terms is consistent
The infinite geometric series is represented using the common ratio to illustrate the ongoing relationship between successive terms
The infinite geometric series can be expressed by the formula \(a + ar + ar^2 + ar^3 + \dots\), where \(a\) is the first term and \(r\) is the common ratio
The sequence \(5, 10, 20, 40, \dots\) has a common ratio of \(r = 2\) and its corresponding series is \(5 + 5 \cdot 2 + 5 \cdot 2^2 + 5 \cdot 2^3 + \dots\)
The formula is essential for understanding the progression of the series and for identifying subsequent terms
The value of the common ratio \(r\) is pivotal in determining whether an infinite geometric series converges or diverges
A series converges to a finite sum when the absolute value of the common ratio is less than one
A series is divergent when the absolute value of the common ratio is equal to or greater than one
The sum of a convergent infinite geometric series can be calculated using the formula \(S = \frac{a}{1-r}\), where \(S\) represents the sum, \(a\) is the first term, and \(r\) is the common ratio
The series \(100, 50, 25, 12.5, \dots\) with a common ratio \(r = \frac{1}{2}\) is convergent, and its sum can be calculated as \(S = 100 \cdot \frac{1}{1 - \frac{1}{2}} = 200\)
Infinite geometric series have real-world applications in finance, physics, and other areas, making the sum formula and convergence criterion important concepts to understand
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First term of an infinite geometric series
Denoted by 'a', it is the initial value from which the series starts.
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Common ratio in an infinite geometric series
Denoted by 'r', it is the factor by which each term is multiplied to get the next term.
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An infinite geometric series will converge if the absolute value of the common ratio, ______, is less than one.
|r|
