 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/discrete-mathematics-probability-and-continuous-mathematics/2/b/v/solution

Treat

a, b

as fixed positive integers. For

n \geq 1

n^{2} + a \leq (1 + a) n^{2}

. Raising this inequality to the positive power

b

gives

(n^{2} + a)^{b} \leq (1 + a)^{b} n^{2 b} .

The factor

(1 + a)^{b}

is independent of

n

, as required by Big O notation. Hence

True for fixed a, b : (n^{2} + a)^{b} = O (n^{2 b}) .

A bound uniform over varying

a, b

is not what the notation here asserts.

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