Repeating Decimals as Geometric SeriesDate: 02/13/2001 at 08:04:14 From: Stephanie Subject: Geometric series with decimals I have no idea how to do this. What are the geometric series of .666... and of 3.5353535...? Could you please help? Thank you, Stephanie Date: 02/13/2001 at 13:47:39 From: Doctor Greenie Subject: Re: Geometric series with decimals Hi, Stephanie - I'm glad to see you are given these as problems with geometric series. Too often, students are taught how to convert repeating decimals to common fractions and then later are taught how to find the sum of infinite geometric series, without being shown the relation between the two processes. Let's do a couple of problems similar to yours using both methods. I will choose the two decimals 0.27272727... 4.16666666... Converting the first decimal to a common fraction by the first method, we have x = 0.27272727... 100x = 27.27272727... and subtracting the two equations we have 99x = 27 x = 27/99 = 3/11 Converting the second fraction by the same process, we have 10x = 41.666666... 100x = 416.666666... and subtracting the two equations we have 90x = 375 x = 375/90 = 75/18 = 25/6 Now for an infinite geometric series we have, when r < 1, a + ar + ar^2 + ar^3 + .... = a / (1-r) Evaluating the repeating decimal 0.27272727... using geometric series, we have 0.272727... = 0.27 + 0.0027 + 0.000027 + 0.00000027 + ... = 0.27 + 0.27(.01) + 0.27(.01)^2 + 0.27(.01)^3 + ... = 0.27 / (1-.01) = 0.27 / 0.99 = 27/99 = 3/11 And evaluating the repeating decimal 4.1666666... using geometric series, we have 4.166666... = 4.1 + .06 + .006 + .0006 + ... = 4.1 + .06 + .06(.1) + .06(.1)^2 + ... = 4.1 + .06 / (1-.1) = 4.1 + .06 / .9 = 4.1 + 6/90 = 4.1 + 1/15 = 41/10 + 1/15 = 123/30 + 2/30 = 125/30 = 25/6 Having written the response above, I just went back and looked at your original question, and it seems I may have answered a bigger question than the one you asked.... - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ |
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