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in limits, what does converges and diverges mean?

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Diverges means that the operation you are performing has no number solution that we can pinpoint (even if we try to do the operation for an infinite amount of years). Converges means that the operation you are performing can be simplified to a number solution (even if it takes infinite years to get the exact answer) If you post some convergent/divergent questions, we can show you how to answer them, or you can look at:
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Diverges is the story of the fat tail. In finance they do calculations, to determine if there is trouble ahead. If it diverges but is negligible, you tell the boss double down on that bet. But if you make a mistake and it has a fat tail, it would come back to bite you in the butt.

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