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Unformatted text preview: (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 + Â·Â·Â· + a n + Â·Â·Â· = âˆž n =1 a n . What does it mean to add up infinitely many numbers? We define the nth partial sum s n of the series by s n = a 1 + a 2 + a 3 + Â·Â·Â· + 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 â†’âˆž s n = lim n â†’âˆž n i =1 a i , if that limit exists. Example. Consider âˆž n =1 1 2 n = 1 2 + 1 4 + 1 8 + Â·Â·Â· + 1 2 n + Â·Â·Â· . 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 â†’âˆž s n = lim n â†’âˆž 1 1 2 n = 1 0 = 1 . Why is s n = 1 1 2 n ? Hereâ€™s a simple way of verifying this: Write s n = 1 2 + 1 4 + 1 8 + Â·Â·Â· + 1 2 n so 2 s n = 1 + 1 2 + 1 4 + Â·Â·Â· + 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 ....
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 Fall '08
 VALDIMARSSON

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