Add infinitely many terms and sometimes get a finite answer: when the ratio sits between -1 and 1, a geometric series converges to S = a/(1-r). It is the cleanest doorway to infinity in the whole course. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.
Two questions decide everything in this topic: does the pattern add the same amount each step, or multiply by the same factor? Add, and you have an arithmetic sequence — linear in n, growth that never changes pace. Multiply, and you have a geometric one — exponential in n. And if that factor is smaller than 1, infinitely many terms can still add up to a finite number.
When the common ratio r satisfies -1 < r < 1, an infinite geometric series converges to the limiting sum S = a/(1-r), where a is the first term, allowing infinitely many terms to sum to a finite value.
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