Taylor series in haskell

Question

How to write Taylor series recursion data like this one:

fib = 0 : scanl (+) 1 fib

For example, I want promt something like this:

fac n = product[1..n]
sin' x = x : x^3/fac(3) : x^5/fac(5) : ...
sum $ take 10 (sin' (pi/6))

And get sum of 10 first elements of sine Taylor series.

Answer 1

That's not quite the Taylor series for sin! But here's a clue ...

products = scanl (*) 1 [1..]
powers x = map (x^) [0..]
exp' x = zipWith (/) (powers x) products

*Main> sum (take 10 (exp' 1))
2.7182815255731922

Answer 2

A common approach is to use a list to represent just the coefficients of the power series and write another function to evaluate the power series at a particular value.

For example, the power series 1/(1-x)^2 = 1 + 2x + 3x^2 + ... would be represented by the list [1,2,3,...].

To evaluate it, we need a function:

eval :: [Float] -> Float -> Float

The base case is easy:

eval [] x = 0

The inductive case can be defined recursively:

eval (a:as) x = a + x*...   -- left as an exercise

Of course, this eval won't terminate on an infinite list, so you'll have make sure to only give it finite lists:

sin' = [ 1, 0, - 1 / fac 3, 0, 1 / fac 5, 0, ... ]
eval (take 10 sin') (pi/6)

Now you just have to generate the coefficients for sin'. Note that in this case 10 refers to the highest power of x you want to sum to, not the number of non-zero terms in the sum.