Write down and simplify the first four terms, in ascending power of $x$, in the expansion of \((8-

Question

Write down and simplify the first four terms, in ascending power of $x$, in the expansion of $(8-9x)^{\frac{1}{3}}$. Hence, by finding a suitable value of $x$, find a rational approximation to $\sqrt[3]{-10}$. Why is this not a good approximation?

Accepted Answer

$(8-9x)^{\frac13}=2-\dfrac{3x}{4}-\dfrac{9x^2}{32}-\dfrac{45x^3}{256}+\cdots$. At $x=2$, the formal estimate is $-\dfrac{65}{32}$; it is unreliable because $2>\dfrac89$, outside the convergence range.

Example involving Binomial Series

Example involving Binomial Series

Solution:

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