Calculus: Analytic Geometry and Calculus, with Vectors

626 Hint: Start by writing 1+2x 1-x-x2 = ao + a1x + a2 + a3X3 + and 1 + 2x = ao + alx + a2x2 + a3x3 + aax4 + -aox-aix2-aax3-a3x4 ...

12.4 Power series 627 For the case in which Fo = Fl = 1 as in (1), the formulas (3) and (6) and (9) show that (10) F. = (Bn+l + ...

628 Series 20 Without pretending to give a reasonable introduction to complex numbers, we take a hasty look at some remarkable f ...

12.4 Power series 629 and it is worthwhile to be able to start with a clean sheet of paper and write all of the formulas needed ...

(^630) Series Let e > 0. Choose an integer N such that Isk - s4 < e/2 when n > N N let C = I Isk - s4. Then, when 0 < ...

12.4 Power series 631 Making wholesale use of the rule of Problem 8 for multiplication of series, show that (5) (^111) °° * 1 1- ...

632 Series when jxj < R. Persons willing to go fishing even when no fish are caught can try to prove the result with rudiment ...

12.5 Taylor formulas with remainders 633 Integrating by parts with u=f'(t) dv=dt du = f"(t) dt, v = - (x - t) gives or f(x) = f( ...

(^634) Series is valid. The right member of (12.56) is called the Taylor expansion of f in powers of (x - a). Problem 5 gives an ...

12.5 Taylor formulas with remainders 635 and use the integral form (12.54) to obtain (12.581) R..(x) =q(q - 1)(q - 2)... (q-n) n ...

636 3 Write two more terms of each series appearing in the calculations rr/2 K - 1 dB J 0 1/1 - k2 sin2 B + (-x) .+. 1 2-) =1+Zx ...

12.5 Taylor formulas with remainders 637 6 Little things like the formula (1) f(1) = f(0) (1 - 0) +LEO)(1 - 0)2 f(n)(0) f(n+I)(1 ...

(^638) Series (7) can be put in the form (9) fck)(t) =[(x- xo)a + (Y- yo) ay] G when k = 2. It turns outthatTaylor formulas with ...

12.5 Taylor formulas with remainders 639 In case AC - B2 > 0, the quantity in braces will be nonnegative whenever jhi and Jkl ...

(^640) Series takes extreme values. 4ns.: The critical points where the first-order partial derivatives both vanish are (0,0) an ...

12.6 Euler-Maclaurin summation formulas 641 B. are defined by the formulas (12.63) B = n!B (0) (n > 2) and (12.631) Bo=1,B1= ...

642 Series which is used to estimate sums. More Euler-Maclaurin formulas (12.661) f(k) =f f(x) dx +f(P) 2 f(q) +f (g) 12f'(P) k= ...

12.6 Euler-Maclaurin summation formulas 643 In many practical applications, the values of the integrals (12.67) f 2 f (m) (x)Bm( ...

644 Series involving binomial coefficients. For example, putting s = 3 gives n k3 = n3 *[n4(1) + 4n3(-C + 6n2(+) + 4n(0)], k=1 s ...

12.6 Euler-Maclaurin summation formulas 645 Remark: We proceed to show how this formula can be used to derive % ery impor- tant ...