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Q1.8 Properties of O and o. For the limit
(a) given that f (x) = O[g(x)], F(x) = O[G(x)], show that fF = O(g G);
(b) given that f (x) = O[g(x)], F(x) = o[G(x)], show that f F = o(g G).
Q1.9 Asymptotic sequences. Verify that these are asymptotic sequences, where n =
0, 1, 2, . . . .
(a) (b)
(c)
Q1.10 Non-uniqueness of asymptotic expansions. Find two asymptotic expansions of the
function valid as based on the asymptotic sequences:
(a) for a suitable function which is to be deter-
mined.
Q1.11 Error function. The error function is defined by

where as Obtain asymptotic expansions of erf(x) for:
(a) and in each case find the general term and decide
if these series are convergent or divergent. Use your result in (b) to give an
estimate for the value of erf(2).
Q1.12 A sine integral. Obtain an asymptotic expansion of

for and decide if the series is convergent or divergent. Also obtain
an expression for the remainder, as an integral, and find an estimate for it.
Q1.13 Exponential integral. Find an asymptotic expansion of

as this will involve Euler’s constant

It might be helpful, first, to show that