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210 7. Approximations and Inequalities

14. Investigate finding an approximate value of fi from a table of powers
    of 2 with integer exponents.


15. Find the polynomial of least degree whose values at -3, -2, -1, 0,
    1, 2, 3, are, respectively, 5, 6, 13, 17, 21, 23, 29.

(a) as the Lagrange polynomial for these data

(b) using the difference operator.

Are your answers to (a) and (b) the same. Why?

16. Factorial powers. Define the rth factorial power oft by

t(O) = 1

t(l) = t

t(‘) = t(t - l)(t -2)e..(t -r+ 1) (rfactors) (r 2 1).

(a) Verify that

t2 = t(2) + t

t3 = tC3) + 30 + t.

(b) Express t4 and t5 in terms of factorial powers.

(c) Show that every (ordinary) power oft, and therefore every poly-

nomial in t, can be expressed as a linear combination

c c&t(‘)

of factorial powers. To see how this can be done systematically,

consult Exploration E.18 in Chapter 2.

(d) Show that, for each r = 1,2,... , At(‘) = rt(‘-l).

17. (a) Show that for any polynomial f(t) of degree n, A”f (t) # 0
    and Akf(t) = 0 for k 2 n + 1. Is the converse true, i.e. if
    An+lf (t) = 0 for some function f(t), must f(t) be a polynomial
    of degree n?
    (b) Deduce from (a), that, for any positive base b # 1, b’ is not a
    polynomial in t.
    (c) Use a difference table to argue that the nth term of the Fibonacci
    sequence defined in Exploration E.14 is not a polynomial in n.


18. (a) Let f(t) be a polynomial of degree not exceeding k over C. Verify,
    that

f(t) = &)k-’ ( “i ) ( t r, 1; l ) f(i)

i=o