series - (12.2 Series A series is the sum of a sequence...

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(12.2) Series A series is the sum of a sequence. If { a n } is a sequence, we may add it up as a 1 + a 2 + a 3 + · · · + a n + · · · = n =1 a n . What does it mean to add up infinitely many numbers? We define the n -th partial sum s n of the series by s n = a 1 + a 2 + a 3 + · · · + a n = n i =1 a i . We say that the series converges if the sequence { s n } of partial sums converges; otherwise, we say the series diverges. That is, the sum of the series is lim n →∞ s n = lim n →∞ n i =1 a i , if that limit exists. Example. Consider n =1 1 2 n = 1 2 + 1 4 + 1 8 + · · · + 1 2 n + · · · . The first four partial sums are s 1 = 1 2 , s 2 = 3 4 , s 3 = 7 8 and s 4 = 15 16 . It turns out s n = 1 - 1 2 n . So the sum of the series is lim n →∞ s n = lim n →∞ 1 - 1 2 n = 1 - 0 = 1 . Why is s n = 1 - 1 2 n ? Here’s a simple way of verifying this: Write s n = 1 2 + 1 4 + 1 8 + · · · + 1 2 n so 2 s n = 1 + 1 2 + 1 4 + · · · + 1 2 n - 1 so subtract to get 2 s n - s n = 1 - 1 2 n (things cancel) so s n = 1 - 1 2 n . Example. A geometric series is any series of the form n =0 ar n = a + ar + ar 2 + ar 3 + · · · , where a and r are constants. We call a the initial term and r the ratio of the series. The
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