The sum to infinity is \( S_\infty = \frac{a}{1 - r} \), and the condition \( |r| < 1 \) is crucial because it ensures the terms get progressively smaller, allowing the series to converge to a finite value. Honestly, I've solved this formula so many times for JEE practice that it feels as automatic as checking the sky before heading to my workshop. But the real reason it sticks is because it mirrors my saving for the terrace repair—small, regular additions from my jute sales, where the total sum approaches a fixed goal, unlike if my expenses (my 'r') were greater than my savings, which would just blow the budget to infinity, yaar.
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