In fact, any series whose general terms a n do not tend to zero will diverge. More precisely, the partial sums are unbounded. That makes sense, right? If you keep adding larger numbers, the running total just gets bigger and bigger. The series of all natural numbers (counting numbers) clearly diverges to infinity. Here are some easy examples to get you started. Next would be s 5 having five terms, and so on.īy definition, the series Σ a n converges to a sum S if and only if the sequence of partial sums converges to S. Of course, the list of partial sums goes on forever. So for example, the first four partial sums of a series are: The kth partial sum for a series Σ a n is the sum of the first k terms of the series: The precise definition for convergence of a series has to do with its partial sums. Here, we would say the series diverges (but not to ∞ nor -∞). As you add term after term, the value of the sum keeps jumping around or oscillating among multiple values. In that case, we say that the series diverges to negative infinity (-∞)
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