Given f(x) = sinh(2x)

(i) Find the Taylor series associated with the function f(x) at x=0.

I found the Taylor polynomials for this function but how do I convert it do Taylor series?

(ii) Use the ratio test to determine for reach values of x this Taylor series is a convergent series.

(iii) By analysing Lagrange's remainder term, show that the Taylor series represent f(x) exactly in the interval you found in the part (b) above.

Do we do this by proving the remainder is equivalent to 0, hence representing f(x)?

Thank you.