Practice section 1 of 7

By considering the power series expansion of $\ln(1+x)$, find the power series representation for $\ln(2+x^2)$ up to and including the term in $x^6$.

Determine the range of x for which the power series expansion of $\ln(2+x^2)$ in Example 1 is valid.

Expand the following in ascending powers of x. Find the range of values of x for which the series expansion is valid.

(i) $(8+x)^{1/3}$, up to and including the term in $x^2$.

(ii) $(1-1/x)^{-1}$, up to and including the term in $x^4$.

Expand the following as a series in ascending powers of x, up to and including the term in $x^3$, stating the set of values of x for which the expansion is valid.

(a) $(4+x)^4$

(b) $(1/x+1)^{-2}$

(c) $(x+4)^{-3/2}$

Express $f(x)=\dfrac{3x+2}{(x-1)(x+4)}$ in partial fractions and hence give the first four terms in the expansion of f(x) in ascending powers of x.

State the range of values of x for which the expansion is valid.

(i) Expand $\sqrt[3]{\dfrac{x+2}{x-1}}$ in ascending powers of x, up to and including the term in $x^2$.

(ii) Find the range of values of x for which the expansion is valid.

(iii) By putting $x=8/11$ into the expansion found in (i), find an approximate value of $\sqrt[3]5$ correct to 4 decimal places.

(iv) Two students suggested alternative substitutions to find an approximate value of $\sqrt[3]5$.

Without performing any calculations, identify the student who made an unacceptable estimate. Explain your answer.

(v) Use the substitution suggested by the other student to find another estimate for $\sqrt[3]5$ correct to 4 decimal places. Explain why this new approximation is better.