+0  
 
0
114
1
avatar+102 

Compute the sum as a common fraction: \(\begin{array}{r r@{}c@{}l} & 1 &.& 11111111\ldots \\ & 0 &.& 11111111\ldots \\ & 0 &.& 01111111\ldots \\ & 0 &.& 00111111\ldots \\ & 0 &.& 00011111\ldots \\ & 0 &.& 00001111\ldots \\ & 0 &.& 00000111\ldots \\ + &&\vdots \\ \hline &&& ~~~? \end{array}\)

 Feb 10, 2020
 #1
avatar+111389 
+1

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)

 

 

cool cool cool

 Feb 10, 2020

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