210 7. Approximations and Inequalities
- Investigate finding an approximate value of fi from a table of powers
of 2 with integer exponents. - 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?
- 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).
- (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. - (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