By M. Peski n, D. Schroeder [RUSSIAN]

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5! 6! h2 h3 h1 f (x0 + h) = f (x0 ) + f (x0 ) + f (x0 ) + f (x0 ) 1! 2! 3! h5 h6 h4 (5) (6) (4) + f (x0 ) + f (x0 ) + f (η3 ) , 4! 5! 6! h1 h2 h3 f (x0 − h) = f (x0 ) − f (x0 ) + f (x0 ) − f (x0 ) 1! 2! 3! 4 5 h h h6 + f (4) (x0 ) − f (5) (x0 ) + f (6) (η4 ) , 4! 5! 6! f (x0 + 2h) = f (x0 ) + f (x0 ) ✰✺❞❆✹ ✖✙✢❊✳✵✩❘✩✥✚✜❂✜✩●✤❅✹❨✤❅✚✸✤❊✹✛✰✣✮❃✦✙✔✣✽ ✮✜✩ ✙✴ ◗ ✶✜⑥❬✶❍✽✸✮✙✰✣✚✜✩❝✰✲❂✸✤❊✽ ★✪✮✙✰✣✮✿✩ ✵ η1 ∈ ]x0 , x0 + 2h[ η2 ∈ ]x0 − 2h, x0 [ η3 ∈ ]x0 , x0 + h[ ★✯✤ η4 ∈ ]x0 − h, x0 [ ◗ δh4 f (x0 ) = f (4) (x0 )h4 ✷✰✲✹❨✤ ✄ + h0 > 0 ♠❴✰✣✚✜✢❝✤❊✰✣✚✙✤ 4 64 (6) f (η1 ) + f (6) (η2 ) − f (6) (η3 ) + f (6) (η4 ) 6!

M 0 m dx hm pm (x) = f (x0 ) + (ii) ✵ ✴✙◗ ✴▲✷✸✶ ✵ ✴✙◗ ✴❏✺✺✶ ❆✾ ✰✣✚✜✩❉✤❅★✯✢❅✱✴✹❁✮✙✰✲✮✜✩❘✾❍★✯✤✥✤❅★❇✩✥★✵✾s✤❊✹✛✰✣✮✈✖✜✔✣✢✫✦✸★✯✚✸■✈✢❊★✪✱✗✔▲✢❀✺✲✚✜★✪✩✪◗ ✖❛❵❛✰✣✜✛★❍✚ ✤✚✢❖ ✰✣✱✲✮✿✏✓✩❴❋❊✯ ✚✸❉❯✤❊✜✐✹✛✘❁✹✻✿❡✩❊★✯❲✫✢❵✍ ✘✛★❝✄✷✹❵✤❊✬✙✘✻✔r✭✪✰✣❬❭✢❅✰✣✳✯✮✜✱✴✾❍✤❊★✧✹✛✰✣⑥✲✮ ◗❁⑥❝✖❛✰✣★✪✚✙✩✥✢❵✤ ✭✯✤❅✔✣❂✙✘❁✹✛✢❵✘❲❏ ★✵❬❭✩●✰✲✤❅✹✛✹❁✩▼✱✗✾✯✔❯✰✣✤❅✮✼✹❁✰✲✤❊✮❚✹✛✮☎✦❋✄✙✱✴❏ ★✪✢❊★✯✢❅✮✼★✯✤❇✚✙✢❵✦✸✭✯✩❊✚✙✢❅✹ ✹ ❖ ❖ ✔✣✔✣❂✙✮✼✘❁✤❊★✲★❝✽❩★✯✮✙✮✼✰✣✤❊✚✜✢❅★✩ f f (m + 1) pm max |f (x) − pm (x)| ≤ 1 hm+1 max |f (m+1) (x)|. 2(m + 1) x∈[x0 ,x0 +mh] ✰✲✫❂✸★✯✤❊✱✗★✪✮✷✔▲✚✈✢❀✺✼✖✿✚✙✔▲★✯✢✫✢❇✦✸✘❲❏✭ ✔▲❖ ✮✜★✯✔✣✘✛✰✣✘❁✰✲✖✜❄✣✖✿✹✛★❚★✪✱❚★✪✮✼★✪✮✼✤❊✢❅✤❘★✗✦✙✘❁★❣★q❸❦✖❛✔❫✰✣✶✷✘✛✶✷✘❁✰✲✮✙✢❘t✣✦✸✱✴★ ★q✦✸★✔✣✚✸✾❆✤❊✰✲★✳✚✙✲❝✢✫✤❅✰✣✦✸✮ ★ ✵ ✴✙◗ ✴▲◗ ⑧✼✶❢★❍✤♥✘❁★✗✖❛✰✣✘✛✶✷✮✙t✣✱✴★ ❵❛✦✜✔▲✜✛✮✜✚✢✩▼✱✲✘✛★✴✏✓❋❊✤❊✬✜❉❯✭✯✜✐✰✣✢❅✿❡✳✯❲ ✱✴✿ ★ ✇①✘✸✴✸★✵◗ ✺✙✩●✤❈✽❑✱✗✖❛✰✲✔✣✩❅✹✛✩✥✩❇✹✛❂✙✔ ✘✛★❘★✪✦❑✾♥❏ ✭✯✘❁★✵✤❅✩▼✔✣❂✙✰✣✘❁✖❛✹✛✢❈✭✯✦✸✢❀✔❯★✵✤❅✩❴★✯✢❊✚✙✭✵✢❀✩✥✩ ✚✙✘❁✤❅✔▲✤❅✩❵★❍✩❊✤ ★✯✱♥❂✙✘✛◗❋✔✣♠❵❂✙✘✛✔▲★✪✢▼✩❴★✯❱❣■✸✾❍★✯★✯✱✴✚✙■♥✖✙✘✛✭✯★✣✮✜✽❋✰✣✹✛✘❈✮✜✾✯★✪✭✪✩✥✤✩ ❬ ✔✲✾❍★✯✹✛✮✈✘❁★♥✘✛★✪✦✸✩❘★▼✖❛✱✴✰✣✰✣✹✛✮✼✮✼✤❅✤❊✩ ✢❅★✯✢❆✺✼✚✙★❇✽✩❊✹ ★✯★✪✤ ✩✥✤✫❖ ✘✛★❇✖✿✰✲✽✿✘❁✶✷✔▲✮✙✘✛✰✣t✲✢❀✱✴✩ ★❇✦✸★❇✦✸★✯❄✲✢❊✭✥✴✴✺✼✚✙✹✕✹❁✮✼✤❅★✯✢❅✖✿✰✲✘❁★▼✘✻✔❚❬❭✰✣✮✜✾❍✤❊✹✛✰✣✮ x∈[x0 ,x0 +mh] ✟ pm x = x0 f q2 x0 − h x0 f ∇m h δhm x0 + h δh2 f (x0 ) f (x0 + h) − 2f (x0 ) + f (x0 − h) d2 q (x ) = = .

5! 6! 1 2 3 (2h) (2h) (2h) f (x0 − 2h) = f (x0 ) − f (x0 ) + f (x0 ) − f (x0 ) 1! 2! 3! (2h)4 (2h)5 (2h)6 (4) (5) (6) + f (x0 ) − f (x0 ) + f (η2 ) , 4! 5! 6! h2 h3 h1 f (x0 + h) = f (x0 ) + f (x0 ) + f (x0 ) + f (x0 ) 1! 2! 3! h5 h6 h4 (5) (6) (4) + f (x0 ) + f (x0 ) + f (η3 ) , 4! 5! 6! h1 h2 h3 f (x0 − h) = f (x0 ) − f (x0 ) + f (x0 ) − f (x0 ) 1! 2! 3! 4 5 h h h6 + f (4) (x0 ) − f (5) (x0 ) + f (6) (η4 ) , 4! 5! 6! f (x0 + 2h) = f (x0 ) + f (x0 ) ✰✺❞❆✹ ✖✙✢❊✳✵✩❘✩✥✚✜❂✜✩●✤❅✹❨✤❅✚✸✤❊✹✛✰✣✮❃✦✙✔✣✽ ✮✜✩ ✙✴ ◗ ✶✜⑥❬✶❍✽✸✮✙✰✣✚✜✩❝✰✲❂✸✤❊✽ ★✪✮✙✰✣✮✿✩ ✵ η1 ∈ ]x0 , x0 + 2h[ η2 ∈ ]x0 − 2h, x0 [ η3 ∈ ]x0 , x0 + h[ ★✯✤ η4 ∈ ]x0 − h, x0 [ ◗ δh4 f (x0 ) = f (4) (x0 )h4 ✷✰✲✹❨✤ ✄ + h0 > 0 ♠❴✰✣✚✜✢❝✤❊✰✣✚✙✤ 4 64 (6) f (η1 ) + f (6) (η2 ) − f (6) (η3 ) + f (6) (η4 ) 6!

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