Adding infinitely many numbers together sounds like it should produce an infinitely large result, or at least an undefined one. For a specific, important category of infinite series called geometric series, where each term is a fixed fraction of the one before it, that intuition is simply wrong: adding one-half, then one-quarter, then one-eighth, and so on forever, produces a sum that gets closer and closer to exactly one, and mathematically never exceeds it — a genuinely counterintuitive but rigorously provable result.
Each partial sum gets closer to the limit without ever reaching or passing it
Consider adding the terms of this series one at a time: one-half alone is one-half; adding one-quarter brings the running total to three-quarters; adding one-eighth brings it to seven-eighths; adding one-sixteenth brings it to fifteen-sixteenths. Each successive partial sum gets closer to one, and the gap remaining between the current partial sum and exactly one is always precisely equal to the next term about to be added — meaning the sum keeps approaching one but never actually reaches or exceeds it at any finite number of terms. What makes the infinite version of this sum genuinely equal to exactly one, rather than merely approaching it forever without reaching it, is a matter of how mathematicians formally define the sum of an infinite series in the first place: the sum is defined as the limit that the sequence of partial sums approaches, and in this specific case, that limit is exactly one.
A general formula covers every geometric series with a small enough ratio
This specific example is one case of a more general result: any geometric series where each term equals the previous term multiplied by some fixed ratio smaller than one in absolute value converges to a finite sum, calculable directly from the series' first term and its common ratio using a simple formula. If the ratio between successive terms is one or larger, however, the series instead grows without bound and has no finite sum at all — the specific value of that ratio is what determines whether an infinite geometric series converges to a tidy finite number or diverges toward infinity, a clean dividing line that makes this one of the more elegant and thoroughly understood results in the broader study of infinite series.
What we're still unsure about
The convergence of geometric series with a ratio smaller than one in absolute value, and the formula for calculating their exact sum, are rigorously proven results in mathematical analysis, established with complete certainty and taught consistently across mathematics curricula. What occasionally trips up students, and is more a matter of building solid intuition than an open mathematical question, is reconciling the everyday sense that "adding infinitely many positive numbers" should surely produce something infinitely large with the precise, formal definition of an infinite sum as a limit of partial sums — the mathematics itself isn't in doubt, but getting comfortable with exactly why that formal definition is the right way to make sense of an infinite sum takes many students real time and practice to internalise.
This sits inside Sequences & Series, one of eight topics in Algebra, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.