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[SOLVED] decimal expansion to rational number

dwsmith

Well-known member
Feb 1, 2012
1,673
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?
 

CaptainBlack

Well-known member
Jan 26, 2012
890
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
 

chisigma

Well-known member
Feb 13, 2012
1,704
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

$\chi$ $\sigma$