Question

Find the sum to n terms of the series 0.6+0.66+0.666+.....................

Answer 1

We have

`0.6+0.66+0.666+0.6666`

Let's take 6 out from the series

`6(0.1+0.11+0.111+0.1111+....`

Now multiply and divide by 9

`(6)/(9)(0.9+0.99+0.999+0.9999+....`


`(6)/(9)(1-0.1+1-0.01+1-0.001+1-0.0001+.... \\ \\(6)/(9)(1-10^(-1)+1-10^(-2)+1-10^(-3)+...`

all 1s add up to n

`(6)/(9)(n-(1)/(10)* (1-((1)/(10))^(n+1))/(1-(1)/(10)))`