- Prove that
where
as
.
- Using part (a) prove
- Proof. We start with the algebraic identity
Replacing
by
we have,
By the definition of
we have
Therefore,
Hence,
- Proof. First, we use the expansion of
,
where
. By part (a) we then have as
,
Next, since
we multiply this expansion for
by the expansion (page 287 of Apostol) for
as
,
I think there is a mistake in the 1/cos(x) line. You didn’t write o(x^5) at the end and even if you did, when expanding you get x^4/4 as the lowest power term and it’s not little oh of x^4, so it reduces to o(x^3) which is gonna ruin the rest of the answer.