Use series to find error or validity

A question is this type if and only if it asks about the error in a series approximation, percentage error, or range of validity of a series.

5 questions

Use a Maclaurin series to find a quadratic approximation for $\ln ( 1 + 2 x )$.
Find the percentage error in using the approximation in part (a) to calculate $\ln ( 1.2 )$.
Jane uses the Maclaurin series in part (a) to try to calculate an approximation for $\ln 3$. Explain whether her method is valid.

Use the Maclaurin series for $\ln ( 1 + x )$ up to the term in $x ^ { 3 }$ to obtain an approximation to $\ln 1.5$.
(A) Find the error in the approximation in part (i).
(B) Explain why the Maclaurin series in part (i), with $x = 2$, should not be used to find an approximation to $\ln 3$.
Find a cubic approximation to $\ln \left( \frac { 1 + x } { 1 - x } \right)$.
(A) Use the approximation in part (iii) to find approximations to
- ln 1.5 and
- $\quad \ln 3$.
  (B) Comment on your answers to part (iv) (A).

Use Maclaurin's series expansion for $\ln ( 1 + x )$ to show that the first three terms of the Maclaurin's series expansion of $\ln ( 1 + 3 x )$ are $$3 x - \frac { 9 } { 2 } x ^ { 2 } + 9 x ^ { 3 }$$ 15
Julia attempts to use the series expansion found in part (a) to find an approximation for $\ln 4$ Julia's incorrect working is shown below. $$\begin{array} { r } \text { Let } 1 + 3 x = 4
3 x = 3
x = 1 \end{array}$$ $$\text { So } \begin{aligned} \ln 4 & \approx 3 \times 1 - \frac { 9 } { 2 } \times 1 ^ { 2 } + 9 \times 1 ^ { 3 }
& \approx 3 - 4.5 + 9
& \approx 7.5 \end{aligned}$$ Explain the error in Julia's working.
15
Use $x = - \frac { 1 } { 6 }$ in the series expansion found in part (a) to find an approximation for $\ln 4$ Fully justify your answer.