Suppose the geometric series \(x+x^2+x^3+\ \cdots\) has a limiting sum.
By considering the graph \(y=-1-\dfrac{1}{x-1}\), or otherwise, find the range of possible values of \(S\). (3 marks)
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\(x + x^2+x^3\ \cdots \Rightarrow a=x, r=x\)
\(S=\dfrac{a}{1-r} = \dfrac{x}{1-x}=\dfrac{-(1-x)+1}{1-x} = -1+\dfrac{1}{1-x}=-1-\dfrac{1}{x-1}\)
\(\text{If limiting sum}\ \Rightarrow \abs{r} \lt 1\ \Rightarrow \ \abs{x} \lt 1\)
\(\Rightarrow \text{Possible values of}\ S =\ \text{range of}\ \ y=-1-\dfrac{1}{x-1}\ \text{in domain}\ \abs{x} \lt 1\)
\(\text{Consider}\ \ y=-1-\dfrac{1}{x-1}:\)
\(\text{Asymptote at}\ \ x=1.\)
\begin{array} {|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & -2 & -1 & 0 & 1 & 2 \\
\hline
\rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} & -\dfrac{2}{3} & -\dfrac{1}{2} & \ \ 0\ \ & \infty & -2 \\
\hline
\end{array}
\(\text{As}\ x \rightarrow \infty, \ y \rightarrow -1\)
\(\text{As}\ x \rightarrow -\infty, \ y \rightarrow -1\)
\(\abs{x} \lt 1\ \Rightarrow \ y \in \Big( -\dfrac{1}{2}, \infty\Big ) \)
\(\therefore\ \text{Possible values of}\ S \in \Big( -\dfrac{1}{2}, \infty\Big ) \)
