The geometric series formula is a mathematical expression used to find the sum of the terms in a geometric series, which is a sequence where each term after the first is found by multiplying the previous term by a constant known as the common ratio. This formula is particularly useful in situations involving exponential growth or decay, finance, and computer science, where sequences exhibit multiplicative patterns. The formula can be represented as $$S_n = a \frac{1 - r^n}{1 - r}$$ for finite series or $$S = \frac{a}{1 - r}$$ for infinite series, where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms in the series.
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The geometric series formula applies to both finite and infinite series, with different forms for each case.
If the common ratio 'r' is greater than or equal to 1, the infinite series does not converge and diverges to infinity.
For a finite geometric series, if 'n' is known, you can directly calculate the sum using the formula $$S_n = a \frac{1 - r^n}{1 - r}$$.
In practical applications like finance, geometric series are used to calculate compound interest over time.
Understanding convergence is crucial when working with infinite geometric series since only those with an absolute common ratio less than 1 will yield a finite sum.
Review Questions
How can you determine if a given geometric series converges or diverges?
To determine if a geometric series converges or diverges, you must look at the common ratio 'r'. If the absolute value of 'r' is less than 1 (|r| < 1), then the infinite geometric series converges and you can use the formula $$S = \frac{a}{1 - r}$$ to find its sum. Conversely, if |r| is greater than or equal to 1 (|r| ≥ 1), the series diverges and does not have a finite sum.
Derive the formula for the sum of a finite geometric series using algebraic manipulation.
To derive the formula for the sum of a finite geometric series, denote the sum as $$S_n = a + ar + ar^2 + ... + ar^{n-1}$$. Multiply both sides by 'r': $$rS_n = ar + ar^2 + ... + ar^n$$. Subtract these two equations: $$S_n - rS_n = a - ar^n$$. Factor out $$S_n$$: $$(1 - r)S_n = a(1 - r^n)$$. Finally, divide both sides by (1 - r) to get $$S_n = a \frac{1 - r^n}{1 - r}$$.
Evaluate the sum of an infinite geometric series with a first term of 5 and a common ratio of 1/3. Discuss what this means in terms of convergence.
To evaluate the sum of an infinite geometric series with first term 'a' equal to 5 and common ratio 'r' equal to 1/3, we apply the infinite geometric series formula: $$S = \frac{a}{1 - r}$$. Plugging in the values gives us $$S = \frac{5}{1 - \frac{1}{3}} = \frac{5}{\frac{2}{3}} = 5 \cdot \frac{3}{2} = 7.5$$. This indicates that as we keep adding terms in this series, they will approach a total sum of 7.5, showing that this particular series converges due to its common ratio being less than 1.
The property of a series to approach a specific value as more terms are added, applicable in the context of infinite geometric series when the common ratio's absolute value is less than 1.