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Question
Find the rational number whose decimal expansion is `0.4bar23`.
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Solution
`S_∞ = a/(1-r)`
S = `0.4bar23 => S = 0.4 + 0.023 + 0.00023 + ... ∞`
`S = 0.4 + 23/10^3 + 23/10^5 + ... ∞`
`S = 0.4 + 23/10^3 {1 + 10^-2 + 10^-4 + ... ∞}`
`S = 0.4 + 23/10^3 {1/(1-1/100)}`
`S = 0.4 + 23/10^3 {100/100-1}`
`S = 0.4 + 23/10^3 xx 100/99`
`S = 4/10 + 23/990`
`S = 419/990`
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