>Does anyone know the sum of this infinite sequence?

>7/10 + 8/100 + 9/1000 + 10/10000 ...

>

>I know the formula of it is (6 + n)/(10^n), but i dont know

>how to find the infinite sum of the sequence.

>

>any ideas are ok

>thanx! I'll assume you're aware that the sum of 1/10 + 1/100 + 1/1000... = 1/9 (if not, just think of 0.11111.....)

6/10 + 6/100 + 6/1000... therefore = 6/9 = 2/3. Deduct that from the series you started with, leaving

1/10 + 2/100 + 3/1000....

That equals 1/9 + 1/100 + 2 /1000... which then equals

1/9 + 1/90 + 1/1000 + 2/10000....which equals

1/9 + 1/90 + 1/900 + 1/10000... and so on

Which gives you 1/9 + 1/90 + 1/900....which equals

1/9 * (1 + 1/10 + 1/100...) = 1/9*(1+1/9) = 10/81.

Add back the 2/3 we took out at the beginning and you have

2/3 + 10/81 = (54+10)/81 = 64/81 = (8/9)^2.

Given the neat answer there may be a more elegant way of doing it.