121616949-math.256

# 121616949-math.256 - a n X b n In general the sequence of...

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242 Chapter 10 Sequences and Series So when | x | < 1 the geometric series converges to k/ (1 - x ). When, for example, k = 1 and x = 1 / 2: s n = 1 - (1 / 2) n +1 1 - 1 / 2 = 2 n +1 - 1 2 n = 2 - 1 2 n and X n =0 1 2 n = 1 1 - 1 / 2 = 2 . We began the chapter with the series X n =1 1 2 n , namely, the geometric series without the first term 1. Each partial sum of this series is 1 less than the corresponding partial sum for the geometric series, so of course the limit is also one less than the value of the geometric series, that is, X n =1 1 2 n = 1 . It is not hard to see that the following theorem follows from theorem 10.2 . THEOREM 10.17 Suppose that a n and b n are convergent sequences, and c is a constant. Then 1. X ca n is convergent and X ca n = c X a n 2. X ( a n + b n ) is convergent and X ( a n + b
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Unformatted text preview: a n + X b n . In general, the sequence of partial sums s n is harder to understand and analyze than the sequence of terms a n , and it is diﬃcult to determine whether series converge and if so to what. Sometimes things are relatively simple, starting with the following. THEOREM 10.18 If ∑ a n converges then lim n →∞ a n = 0. Proof. Since ∑ a n converges, lim n →∞ s n = L and lim n →∞ s n-1 = L , because this really says the same thing but “renumbers” the terms. By theorem 10.2 , lim n →∞ ( s n-s n-1 ) = lim n →∞ s n-lim n →∞ s n-1 = L-L = 0 ....
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