+101 Compute the sum as a common fraction: \(\begin{array}{r r@{}c@{}l} & 1 &.& 11111111\ldots \\ & 0 &.& 11111111\ldots \\ & 0 &.& 01111111\ldots \\ & 0 &.& 00111111\ldots \\ & 0 &.& 00011111\ldots \\ & 0 &.& 00001111\ldots \\ & 0 &.& 00000111\ldots \\ + &&\vdots \\ \hline &&& ~~~? \end{array}\)
+129852 This is an infinite geometric sum and can be expressed as
1 + (1/9) + (1/9) + (1/90) + (1/900) + ......... =
(10/9) + (1/9) + (1/90) + (1/900) + .........
The first term is (10/9) and the common ratio = (1/10)
So......the infinite sum is
(10/9) (10/9)
_________ = ________ = (10/9) * (10/9) = 100 / 81
1 - (1/10) (9/10)
