MHB Decimal expansion to rational number

AI Thread Summary
The discussion focuses on converting the decimal expansion 0.334444... into a rational number. The expression is broken down into a finite part, 33 × 10^-2, and an infinite geometric series starting from n=3. Participants suggest summing the infinite series and combining it with the finite part to find the equivalent rational form. The final conclusion is that 0.33444... can be expressed as 301/900. This demonstrates the method of representing repeating decimals as fractions.
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I am given the number $.334444\ldots$

So we have $0+33\times 10^{-2} + \sum\limits_{n=3}^{\infty}4\times 10^{-n}$

Is there a way to put this all together in one geometric series?
 
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dwsmith said:
I am given the number $.334444\ldots$

So we have $0+33\times 10^{-2} + \sum\limits_{n=3}^{\infty}4\times 10^{-n}$

Is there a way to put this all together in one geometric series?

No, you sum the infinite geometric series, and add \(33/100\). Or sum:

\(\sum\limits_{n=1}^{\infty}4\times 10^{-n}=(4/10)(10/9)\)

and subtract \(11/100\)

CB
 
dwsmith said:
I am given the number $.334444\ldots$

So we have $0+33\times 10^{-2} + \sum\limits_{n=3}^{\infty}4\times 10^{-n}$

Is there a way to put this all together in one geometric series?

$\displaystyle .33444...= \frac{1}{3} + \frac{1}{900} = \frac{301}{900}= \frac{1}{100}\ \frac{301}{9}= \frac{301}{1000}\ \sum_{n=0}^{\infty} 10^{-n}$

Kind regards

$\chi$ $\sigma$
 
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