Limits

Show that

Hence determine the Maclaurin series expansion for y, in ascending powers of x, up to and including the term in x², giving each term in simplest form.

Use the series expansion for sinx in ascending powers of x to determine, in simplest form, the first 2 non‑zero terms in ascending powers of x of the series for sin 3 x.

Use the answers to parts (b) and (c) to show that

Use L’Hospital’s Rule to show that

y = x²e^3x

Use Leibnitz’s theorem to show that, fork

where A, B and C are integers to be determined.

Hence determine the values of x for which

Use L’Hospital’s rule to show that

Show that

Hence determine the first 3 non-zero terms in ascending powers of x of the Maclaurin series expansion of ln(e^2x cos 3x), giving each coefficient in simplest form.

Use the Maclaurin series expansion for ln(1 + x) to write down the first 4 non-zero terms in ascending powers of x of the Maclaurin series expansion of ln(1 + kx), where k is a constant.

Hence determine the value of k for which

exists.

The Taylor series expansion of f (x)about x a is given by

Use differentiation to determine the Taylor series expansion of lnx, in ascending powers of (x – 1), up to and including the term in (x – 1)²

Hence prove that

Use L’Hospital’s rule to determine

(Solutions relying entirely on calculator technology are not acceptable.)

The Taylor series expansion of x about x a is given by

Given that

Determine the Taylor series expansion about x = 1 of

in ascending powers of (x – 1), up to and including the term in (x – 1)³

Give each coefficient in simplest form.

Use this series expansion to evaluate

explaining your reasoning clearly.

Use l’Hospital’s Rule to show that