$1 + \frac{1}{r} + \frac{1}{r^2} + \frac{1}{r^3} + \frac{1}{r^4} + \cdots \\ { \LARGE =} \\ 1 - \frac{1}{s} + \frac{1}{s^2} - \frac{1}{s^3} + \frac{1}{s^4} + \cdots \\$

What is the relationship between $r$ and $s$ ?

$r+s = 0$
$r + s = 1$
$rs = 0$
$rs = 1$

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Relevant wiki: Geometric Progression Sum$1+\dfrac{1}{r}+\dfrac{1}{r^2}+\dfrac{1}{r^3}+\dfrac{1}{r^4} + \ldots = 1-\dfrac{1}{s}+\dfrac{1}{s^2}-\dfrac{1}{s^3}+\dfrac{1}{s^4} - \ldots$

Notice that the LHS is the sum of a geometric progression to infinity, where $a=1$ and $r=\dfrac{1}{r}$

Similarly, notice that the RHS is also the sum of another geometric progression to infinity, where $a=1$ and $r=-\dfrac{1}{s}$

Therefore, we can replace these two sides of the equation with the formula for sums of GP to infinity:

$\dfrac{1}{1-\frac{1}{r}} = \dfrac{1}{1-\left(-\frac{1}{s}\right)}\\ \dfrac{1}{\frac{r-1}{r}} = \dfrac{1}{\frac{s+1}{s}}\\ \dfrac{r}{r-1} = \dfrac{s}{s+1}\\ r(s+1)=s(r-1)\\ rs+r=rs-s\\ \boxed{r+s=0}$