Answer
(a) common ratio$: r=x$
(b) $f(x)=\frac{1}{1-x},\ |x| \lt 1$
(c) please see step by step.
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Work Step by Step
(a) common ratio$: r=x$
(b) $1 +x+x^{2}+\displaystyle \cdots=\sum_{n=0}^{\infty}x^{n}=\frac{1}{1-x},\ |x| \lt 1$
(c)
$f_{1}(x)=\displaystyle \frac{1}{1-x}$
$f_{3}(x)=S_{3}=1+x+x^{2}$
$f_{5}(x)=S_{5}=1+x+x^{2}+x^{3}+x^{4}$
In the vicinity of x=1,
the graphs of $f_{3}$ and $f_{5}$ become very close to the graph of $f_{1}.$
