> I imagine bc is using some sort of power series

There is also a cool Haskell package, numbers, that has all arithmetics defined via power series:

  $ cabal v2-repl -b numbers
    ⡇
  λ> import Data.Number.CReal

  λ> cos pi + 1 :: CReal
  0.0

  λ> :set +s
  λ> showCReal 10000 pi
  "3.141592653589793238462643383279502884197169399375105820…
  (2.69 secs, 688,668,864 bytes)