Infinite series (damn mickey mouse)

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SUMMARY

The repeating decimal 0.27272727... can be expressed as an infinite series: 0.27 + 0.0027 + 0.000027 + ... . This series converges due to its common ratio of 0.01, derived from the division of 0.0027 by 0.27. The sum of the series is calculated using the formula S = a / (1 - r), resulting in S = 0.27 / 0.99, which simplifies to 27/99 or 3/11.

PREREQUISITES
  • Understanding of infinite series and convergence
  • Familiarity with geometric series and their properties
  • Knowledge of the formula for the sum of a geometric series
  • Basic arithmetic operations with decimals and fractions
NEXT STEPS
  • Study the properties of geometric series in detail
  • Learn about convergence tests for infinite series
  • Explore decimal representations and their conversions to fractions
  • Practice problems involving infinite series and their sums
USEFUL FOR

Students studying calculus, mathematics enthusiasts, and anyone interested in understanding infinite series and their applications.

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Homework Statement



the repeating decimal . 27272727 . . . can be written as an infinite series. Write it as a series and tell if it diverges/converges. If it converges, find the sum.


Mickey, also a former student, knows how to do this one. Mickey knows enough math to write it as .27 + .0027 + .000027 . . .

It's now an infinite series. Mickey spots the common ratio of the series as .0027/.27 = .01. Therefore, it converges! He use the formula and presto:
S = .27/(1 - .01) = .27/.99 = 27/99 = 3/11
(remarkable achievement considering he's a mouse!)



The Attempt at a Solution



why did mickey wrote .0027 after .27?? why not .027??
 
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because .27 + .0027 + .000027 = .272727
.27 + .027 = .297 -> .297 +.0027 = .2997 etc
 
ahhh..haven't thought of it...thanks
 

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