Question

How do I prove that 0.3333.. Is equal to 1/3 using a geometric series?

Answer 1

Please see explanation.

Explanation:

`0.\overline{3}`

`=(1)/(10)(3)+((1)/(10))^2(3)+((1)/(10))^3(3)+((1)/(10))^4(3)\cdots`

`=3 \cdot (1)/(10)(1+(1)/(10)+((1)/(10))^2+((1)/(10))^3 \cdots`

`=3 \cdot (1)/(10)\sum_(i=1)^(\infty)((1)/(10))^(n-1)`

`=3 \cdot (1)/(10) \cdot (1)/(1-(1)/(10))`

`=(3)/(10) \cdot (10)/(10-1)`

`=(3)/(10) \cdot (10)/(9)`

`=(3)/(9)`

`=(1)/(3)`