A series converges if its sequence of partial sums

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A series converges if its sequence of partial sums con- verges. The sum of the series is the limit of the sequence of partial sums. For example, consider the geometric series defined by the se- quence a n = 1 r n , n = 0 , 1 , 2 ,... Thus k = 0 ( 2 π ) k = and k = 2 ( 2 π ) k = . Another situation in which we we can actually compute the par- tial sums occurs if those sums are collapsing . It may not be obvious that that is the case, but look for it in this example: n k = 0 1 k 2 + 9 k + 20 = and thus k = 0 1 k 2 + 9 k + 20 = .
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