- 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
- 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.