11.2.51-eg-1 - ∞ X n =0 ar n with a = 23 100 and r = 1...

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Example Express the number 0 . 23 = 0 . 232323 . . . as the ratio of two integers. Start by expanding the decimal into a series 0 . 23 = 0 . 232323 ... = 23 100 + 23 10000 + 23 1000000 + ··· = 23 ± 1 10 2 + 1 10 4 + 1 10 6 + ··· ² = 23 100 ± 1 + 1 10 2 + 1 10 4 + ··· ² = 23 100 X n =0 ± 1 10 ² 2 n = 23 100 X n =0 ± 1 100 ² n which is a geometric series of the form
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Unformatted text preview: ∞ X n =0 ar n with a = 23 100 and r = 1 100 . Since | r | < 1, the series converges to a 1-r or . 23 = 23 / 100 1-(1 / 100) = 23 100-1 = 23 99 ....
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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