Moderate -0.8 This is a straightforward application of the binomial expansion formula for fractional powers with standard coefficient calculation and validity condition |2x| < 1. It requires routine recall of the generalized binomial theorem and basic algebraic manipulation, making it easier than average but not trivial due to the fractional power and negative terms in the sequence.
2 Show that \(( 1 + 2 x ) ^ { \frac { 1 } { 3 } } = 1 + \frac { 2 } { 3 } x - \frac { 4 } { 9 } x ^ { 2 } + \ldots\), and find the next term in the expansion.
State the set of values of \(x\) for which the expansion is valid.
2 Show that $( 1 + 2 x ) ^ { \frac { 1 } { 3 } } = 1 + \frac { 2 } { 3 } x - \frac { 4 } { 9 } x ^ { 2 } + \ldots$, and find the next term in the expansion.\\
State the set of values of $x$ for which the expansion is valid.
\hfill \mbox{\textit{OCR MEI C4 2009 Q2 [6]}}