121616949-math.255 - k(1 x x 2 x 3 ·· x n)1-k(1 x x 2 x...

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10.2 Series 241 is a sequence then the associated series is X i =0 a n = a 0 + a 1 + a 2 + · · · Associated with a series is a second sequence, called the sequence of partial sums { s n | n Z 0 } : s n = n X i =0 a n . So s 0 = a 0 , s 1 = a 0 + a 1 , s 2 = a 0 + a 1 + a 2 , . . . A series converges if the sequence of partial sums converges, and otherwise the series diverges. EXAMPLE 10.16 If a n = kx n , X n =0 a n is called a geometric series . A typical partial sum is s n = k + kx + kx 2 + kx 3 + · · · + kx n = k (1 + x + x 2 + x 3 + · · · + x n ) . We note that s n (1 - x ) = k (1 + x + x 2 + x 3 + · · · + x
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Unformatted text preview: k (1 + x + x 2 + x 3 + ··· + x n )1-k (1 + x + x 2 + x 3 + ··· + x n-1 + x n ) x = k (1 + x + x 2 + x 3 + ··· + x n-x-x 2-x 3- ··· -x n-x n +1 ) = k (1-x n +1 ) so s n (1-x ) = k (1-x n +1 ) s n = k 1-x n +1 1-x . If | x | < 1, lim n →∞ x n = 0 so lim n →∞ s n = lim n →∞ k 1-x n +1 1-x = k 1 1-x ....
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