How do you find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

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Answer 1 · 2016-02-13T22:39:49

It will be $0.bar(9)$ which is taken as equal to 1

For instance:

$1/3=0.bar(3)$

$3xx1/3=0.bar(9)$ but is also equal to 1, therefore both numbers are equal.

Answer 2 · 2016-02-13T23:23:46

Apply the geometric series formula for $a=9/10$ and $r=1/10$ to find that $0.9+0.09+0.009+... = 1$

The other answer is correct, but to add how applying the geometric series formula would give the result:

For $|r| < 1$ and $a!=0$ we have $sum_(n=0)^oor^n = a/(1-r)$

(for a short derivation of this formula, see this answer)

Applying this, as $|1/10| < 1$ , we have

$0.9+0.09+0.009+... = sum_(n=0)^oo9/10(1/10)^n$

$=(9/10)/(1-1/10)$

$=(9/10)/(9/10)$

$=1$