Infinite Series

Definition

An infinite series is a mathematical expression that represents the sum of an infinite number of terms, where each term is a function of the index of the series. Infinite series are particularly relevant in the context of perpetuities, which are a type of annuity that continues indefinitely.

5 Must Know Facts For Your Next Test

1. Infinite series are used to model financial instruments that continue indefinitely, such as perpetuities and annuities.
2. The convergence of an infinite series is determined by the behavior of the terms as the index approaches infinity.
3. Geometric series are a special type of infinite series where the ratio between consecutive terms is constant.
4. The present value of an infinite series can be calculated using the formula for the sum of a geometric series.
5. Convergence of an infinite series is crucial for determining the present value of a perpetuity or other infinite cash flow stream.

Review Questions

• Explain how the concept of an infinite series is applied in the context of perpetuities.
  • Perpetuities are financial instruments that provide a continuous stream of cash flows without end. The present value of a perpetuity can be calculated using the formula for the sum of an infinite geometric series, where the first term is the periodic cash flow and the common ratio is the discount rate. This allows for the determination of the lump-sum value of the perpetual cash flows, which is a key concept in the valuation of perpetuities.
• Describe the role of convergence in the analysis of infinite series related to perpetuities.
  • The convergence of an infinite series is crucial in the context of perpetuities because it determines whether the present value of the series is finite or infinite. For a perpetuity to have a well-defined present value, the infinite series representing the cash flows must converge. This means that as the number of terms in the series approaches infinity, the sum of the terms approaches a finite value. Convergence is typically achieved when the discount rate used to value the perpetuity is greater than the growth rate of the cash flows.
• Analyze how the formula for the sum of a geometric series is applied in the valuation of perpetuities.
  • The formula for the sum of a geometric series, S = \frac{a}{1-r}, where a is the first term and r is the common ratio, is directly applicable to the valuation of perpetuities. In the case of a perpetuity, the first term a represents the periodic cash flow, and the common ratio r is the discount rate. By substituting these values into the formula, one can derive the present value of the perpetuity as PV = \frac{C}{r}, where C is the periodic cash flow and r is the discount rate. This formula is a fundamental tool in the analysis and valuation of perpetuities.

Related terms

Convergence: The property of an infinite series where the sum of the terms approaches a finite value as the number of terms increases without bound.

Geometric Series: A type of infinite series where each term is a constant multiple of the previous term, following a geometric progression.

Present Value: The current worth of a future sum of money or stream of cash flows, discounted at an appropriate interest rate.