Demonstrate that you can generate the series for ex as (define exp-series (stream-cons 1 (integrate-series-tail exp-series))) Explain the reasoning behind this definition. Show how to generate the series for sine and cosine, in a similar way, as a pair of mutually recursive definitions.

To demonstrate that we can generate the series for `exp` using the definition `exp-series = (stream-cons 1 (integrate-series-tail exp-series))`, we can define `exp-series` as follows:

(define exp-series (stream-cons 1 (integrate-series-tail exp-series)))

The reasoning behind this definition is based on the power series expansion of `exp(x)`, which is:

exp(x) = 1 + x + (x^2 / 2!) + (x^3 / 3!) + ...

We can represent this series as a stream by defining `exp-series` as the infinite stream of coefficients:

1, 1, 1/2!, 1/3!, 1/4!, ...

The first term is `1`, and the remaining terms can be generated by integrating the series tail, which means dividing each term by its index and adding the result to the next term. This is exactly what the `integrate-series-tail` procedure does.

To generate the series for `sine` and `cosine`, we can use a pair of mutually recursive definitions. The series for `sine` and `cosine` are related by the following differential equations:

d/dx(sin(x)) = cos(x)
d/dx(cos(x)) = -sin(x)

We can represent these equations as a pair of mutually recursive series:

(define sine-series (stream-cons 0 (cosine-series)))
(define cosine-series (stream-cons 1 (stream-map - (integrate-series-tail sine-series))))

The first term of `sine-series` is `0`, and the first term of `cosine-series` is `1`. The remaining terms can be generated by integrating and negating `sine-series` to get `cosine-series`, and integrating and negating `cosine-series` to get `sine-series`. This generates the infinite streams of coefficients for `sine` and `cosine`.