6.042/18.062] Mathematics for Computer Science March 11. 2005 Srini devadas and Eric Lehman Notes for Recitation 10 1+z+z 1+2+x2+…=12 1+2+3+...+n= n(n+1) 12+2+32+,+2n(n+3)(n+1) 13+23+33+..+n3 n2(n+1)2 4 Theorem(Taylor's theorem). Suppose that f R-R is n+ 1 times differentiable on the interval 0, c]. Then ()=10)+r(0x+r"0y2+…+-O) (n+1) (n+1)! for some z∈[0.,x6.042/18.062J Mathematics for Computer Science March 11, 2005 Srini Devadas and Eric Lehman Notes for Recitation 10 n 1 + z + z2 + . . . + zn−1 = 1 − z (z = 1) 1 − z � 1 1 + z + x2 + . . . = 1 − z (|z| < 1) n(n + 1) 1 + 2 + 3 + . . . + n = 2 2 1 12 + 22 + 32 + . . . + n = n(n + 2 )(n + 1) 3 2 n (n + 1)2 3 13 + 23 + 33 + . . . + n = 4 Theorem (Taylor’s theorem). Suppose that f : R R is n + 1 times differentiable on the interval [0, x]. Then → 2 n f(n+1)(z)xn+1 f��(0)x f(n) (0)x f(x) = f(0) + f� (0)x + + . . . + + for some z ∈ [0, x]. 2! n! (n + 1)!