Power series represented as infinite streams
In section 2.5.3 we saw how to implement a polynomial arithmetic system representing polynomials as lists of terms. In a similar way, we can work with power series, such as
represented as infinite streams. We will represent the series a₀ + a₁x + a₂x² + a₃x³ + ··· as the stream whose elements are the coefficients a₀, a₁, a₂, a₃, ....
a. The integral of the series a₀ + a₁x + a₂x² + a₃x³ + ··· is the series
where c is any constant. Define a procedure integrate-series that takes as input a stream a₀, a₁, a₂, ... representing a power series and returns the stream a₀, (1/2)a₁, (1/3)a₂, ... of coefficients of the non-constant terms of the integral of the series. (Since the result has no constant term, it doesn't represent a power series; when we use integrate-series, we will cons on the appropriate constant.)
b. The function x → eˣ is its own derivative. This implies that eˣ and the integral of eˣ are the same series, except for the constant term, which is e⁰ = 1. Accordingly, we can generate the series for eˣ as
(define exp-series (cons-stream 1 (integrate-series exp-series)))
Show how to generate the series for sine and cosine, starting from the facts that the derivative of sine is cosine and the derivative of cosine is the negative of sine:
(define cosine-series (cons-stream 1 <??>)) (define sine-series (cons-stream 0 <??>))
Nobody's finished this exercise yet. You'll be the first!
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