>>> approx_e(2)
(273, 2.713301767821834, 2.723277141968238)
>>> approx_e(3)
(2719, 2.717781945201135, 2.7187818649749396)
>>> approx_e(4)
(27184, 2.718231830478112, 2.7183318279703124)
>>> approx_e(5)
(271829, 2.7182768284461742, 2.7182868284345063)
>>> approx_e(6)
(2718283, 2.7182813290770236, 2.718282329076777)