You stated that in order to have an infinite sum, the series has to be geometric. Wouldn't an arithmetic series also have an infinite sum?

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In order for an infinite geometric series to have a sum, the absolute value of r, (ratio) must be less than one. You would then be adding a progressively smaller and smaller number. In an arithmetic series, you are always adding the same number so your answer will continue to grow in either the positive or negative direction. It has no limit.