 Solution

The coefficient index supplies the multiplier for a formal derivative and the divisor for a zero-constant formal integral:

deriv :: Num a => [a] -> [a]
deriv as = zipWith (*) (map fromInteger [1..]) (tail as)

integral :: Fractional a => [a] -> [a]
integral as = 0 : zipWith (/) as (map fromInteger [1..])

Using fromInteger makes the index stream unbounded before conversion to the coefficient type. The formal derivative drops the original constant coefficient; the formal integral prepends its chosen zero constant. With exact Rational coefficients these are exact algebraic operations. Analytic convergence is unnecessary for operations on formal power series over a characteristic-zero field.