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282 Answers to Exercises and Solutions to Problems

Consider the difference

p(t) -- A = PM - w(t)

q(t) t - al q(t)

where q(t) = (t-ul)ql(t) an d cl is a constant to be determined. The degree

of the numerator is strictly less than the degree of q(t). Since ql(a1) # 0, we

can choose cl so that p(al)-clql(al) = 0. Thusp(t) = clql(t)+(t-al)pl(t),

for some polynomial PI(t).

Hence

P(t> = c1

q(t)

I PlW

t - a1 a(t)

Since degpl(t) < degql(t), th e induction hypothesis can be applied to

m(Wq1(t).

(b) If the sum is put over a common denominator, the numerator is

p(t) = CCi(t - al).~.(tTii)...(t -a,). (*>

(The hat denotes a deleted term.) Note that

q’(t) = C(t - ~1). +. (t Zj)... (t - ura),

SO that p(ai) = ciq’(ai).

cc>

f(t) = k[bi(t - al)... (t Si)... (t - u,)/q’(ui)].

i=l

6.5. (a)

n

c

1 “1 1

c

1 n-l

k=2k(k-l)=k=21i-~=1-~= n ’

(b) When n = 2, the sum is l/6. For n > 3, we have

= ; p29+q

k=l k=2 k=3

= i[l-l/2-l/n+l/(n+l)]

= (n+2)(n- 1)
4n(n+l) ’
(c) If x = 0, th e sum is equal to n - 1. If x = -1, -l/2,... ,-l/n, the
sum is not defined.
n

c k=2 (kc + l)((:- 1)” + 1) = : (k - 1’,x + 1 - kzi (^1 1)