Geometric series

Definition

A geometric series is the sum of the terms of a geometric sequence, which is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This type of series can be finite or infinite, and its properties are essential for understanding convergence and sums in mathematics.

5 Must Know Facts For Your Next Test

1. The formula for the sum of a finite geometric series with 'n' terms, first term 'a', and common ratio 'r' is given by: $$ S_n = a \frac{1 - r^n}{1 - r} $$ when r is not equal to 1.
2. For an infinite geometric series with first term 'a' and common ratio 'r', if \(|r| < 1\), the sum can be calculated as: $$ S = \frac{a}{1 - r} $$.
3. Geometric series are used in various applications, including finance for calculating the present value of annuities and in computer science for analyzing algorithms.
4. The convergence of an infinite geometric series depends entirely on the value of the common ratio; if it is outside the range (-1, 1), the series diverges.
5. When dealing with geometric series, recognizing whether it is finite or infinite is crucial for applying the correct summation formulas.

Review Questions

• How do you derive the formula for the sum of a finite geometric series?
  • To derive the formula for the sum of a finite geometric series, consider the sum represented as S = a + ar + ar^2 + ... + ar^{n-1}. Multiplying both sides by the common ratio r gives rS = ar + ar^2 + ... + ar^n. By subtracting these two equations, we have S - rS = a - ar^n. Factoring out S results in S(1 - r) = a(1 - r^n), leading to S = a(1 - r^n)/(1 - r) when r is not equal to 1.
• What is the significance of the common ratio in determining whether an infinite geometric series converges?
  • The common ratio plays a critical role in determining convergence for an infinite geometric series. If the absolute value of the common ratio is less than one (|r| < 1), the terms get smaller and approach zero, allowing the series to converge to a specific sum. Conversely, if |r| is greater than or equal to one, the terms do not approach zero and thus the series diverges, indicating it does not settle at any finite value.
• Evaluate how understanding geometric series can impact real-world applications such as finance and computer science.
  • Understanding geometric series is vital in fields like finance where it aids in calculating present values of cash flows over time using annuities. In computer science, it helps analyze algorithms where resources may grow exponentially, such as data storage requirements. Both contexts rely on accurately summing terms in geometric series to make informed decisions regarding investments or system efficiencies, showing how foundational math concepts directly influence practical scenarios.

Related terms

common ratio:

The fixed number that each term in a geometric sequence is multiplied by to obtain the next term.

finite geometric series: A geometric series that has a specific number of terms, allowing for a straightforward calculation of its sum using a formula.

infinite geometric series: A geometric series that continues indefinitely, which can converge to a limit if the absolute value of the common ratio is less than one.