Into the largest and smallest accumulation points of

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into the largest and smallest accumulation points of { r a n } . If r < 0 then the multiplication transfers the largest accumulation point of { a n } into the smallest accumulation point of { r a n } , and the other way round. Warning. It may well happen that lim sup( a n + b n ) 6 = lim sup a n + lim sup b n and/or lim inf( a n + b n ) 6 = lim inf a n + lim inf b n . 1
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SERIES Series is just an infinite sum of the form n =1 a n or a 1 + a 2 + a 3 + . . . , where a n R . Definition. We say that the series converges and write n =1 a n = c if the se- quence of finite partial sums s k = k n =1 a n converges to c . Equivalently ε > 0 k ε : k > k ε k X n =1 a n - c < ε. We say that a series is divergent if it is not convergent. Example. The series 1 - 1 + 1 - 1 + 1 . . . does not converge because its partial sums oscillate between -1 and 1. Example. An infinite decimal fraction is an example of a convergent series. In- deed, when we write a = 0 .a 1 a 2 a 3 . . . we actually mean that a = n =1 a n 10 - n . Remark. The series n =1 a n converges if and only if the series n = m a n con- verges. In other words the first few terms do not affect convergence, even though they change the value of the sum. Indeed, if s k are the partial sums of the original series then the partial sums σ k of the series n = m a n are equal to s k - a where a = m - 1 n =1 a n . From the definition of convergence for sequences, it follows that the sequence { s k } converges if and only if the sequence { σ k } converges. Example (geometric progression). If | b | < 1 then the series n =0 b n converges and n =0 b n = (1 - b ) - 1 . Indeed, by induction in m we obtain m X n =0 b n = (1 - b ) - 1 - b m +1 (1 - b ) - 1 . If | b | < 1 then b m +1 0 and, consequently, m X n =0 b n (1 - b ) - 1 as m → ∞ . Theorem. If the series n =1 a n converges then lim n →∞ a n = 0.
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  • Fall '09
  • Limits, lim, Mathematical analysis, Limit of a sequence, lim sup

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