Maclaurin series and Landau's big "Oh" problem.

I have found the first 5 Taylor polynomials about x=0 for cosh(x). Being p0=1, p1=1, p2=1+ , p3=1+ , p4=1+ + .

How do I Find the Maclaurin series for cosh(x)?

I can see it should be something like

but how can I show a methodic way to get there from the polynomial approximations?

Also, how do I use Landau's big O notation to find

I simplify this with identity to

then change them to Maclaurin approximations with big O error.

How do I simplify that expression to get the limit?

Thank you for any help.