Finding steady states using infinite series Solve Exercise 40 by expressing the amount of aspirin in your blood as a geometric series and evaluating the series.

검증된 단계별 안내

Identify the problem context: The amount of aspirin in the blood over time can be modeled as a geometric series because each dose contributes a fraction of the previous amount due to elimination from the body.

Define the terms of the geometric series: Let the initial dose be \( D \), and let the fraction of aspirin remaining after each time interval be \( r \) (where \( 0 < r < 1 \)). Then the amount after the first dose is \( D \), after the second dose is \( D \times r \), after the third dose is \( D \times r^2 \), and so on.

Write the infinite geometric series representing the total amount of aspirin in the blood as \( S = D + D r + D r^2 + D r^3 + \cdots \).

Recall the formula for the sum of an infinite geometric series when \( |r| < 1 \): \[ S = \frac{a}{1 - r} \] where \( a \) is the first term. In this case, \( a = D \).

Substitute the values into the formula to express the steady-state amount of aspirin in the blood as \( S = \frac{D}{1 - r} \). This represents the long-term amount after many doses.

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주요 개념

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Geometric Series

A geometric series is the sum of the terms of a geometric sequence, where each term is found by multiplying the previous term by a constant ratio. It can be expressed as S = a + ar + ar² + ... , and if |r| < 1, the infinite series converges to S = a / (1 - r). This concept is essential for modeling repeated processes like drug accumulation.

Steady State in Infinite Series

The steady state refers to a condition where the quantity being modeled stabilizes and no longer changes significantly over time. In the context of an infinite geometric series, the steady state corresponds to the sum of the infinite series, representing the long-term amount of aspirin in the blood after many doses.

Modeling Drug Concentration with Series

Drug concentration in the bloodstream over time can be modeled using infinite series by considering repeated doses and elimination rates. Each dose contributes an amount that decreases geometrically due to metabolism, allowing the total concentration to be expressed as a geometric series whose sum predicts the steady-state level.