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Introduction to Infinite Geometric Series

In mathematics, a series is defined as the sum of an infinite sequence of terms. One type of series is the infinite geometric series, which is a sum of an infinite sequence where each term is multiplied by a constant ratio to get the next term. In this article, we will explore the properties and formulas of infinite geometric series.

General Form

The general form of an infinite geometric series is:

∑n=0∞arn\sum_{n=0}^{\infty} ar^n

where a is the first term and r is the common ratio between consecutive terms.

Convergence and Divergence

The convergence or divergence of an infinite geometric series depends on the value of the ratio r. If |r| < 1, the series converges to a finite value as n approaches infinity. If |r| ≥ 1, the series diverges and does not have a finite sum.

Formula for Convergent Geometric Series

If a geometric series is convergent, the sum can be calculated using the following formula:

S=a1−rS = \frac{a}{1-r}

where S is the sum of the series.

Examples

Let's look at some examples of infinite geometric series:

Example 1

∑n=0∞12n\sum_{n=0}^{\infty} \frac{1}{2^n}

In this series, a = 1 and r = 1/2. Since |r| < 1, the series converges. Using the formula, we can find the sum:

S=11−1/2=2S = \frac{1}{1-1/2} = 2

Therefore, the sum of the series is 2.

Example 2

∑n=0∞3n\sum_{n=0}^{\infty} 3^n

In this series, a = 1 and r = 3. Since |r| ≥ 1, the series diverges and does not have a finite sum.

Applications

Infinite geometric series have many applications in mathematics and science. For example, they can be used to model population growth, compound interest, and radioactive decay.

Conclusion

Infinite geometric series are a type of series where each term is multiplied by a constant ratio to get the next term. The convergence or divergence of the series depends on the value of the ratio. If the ratio is less than 1, the series converges to a finite value. If the ratio is greater than or equal to 1, the series diverges. The sum of a convergent geometric series can be calculated using a formula. Infinite geometric series have many applications in mathematics and science.

無限等比級数[JA]