Let's determine which geometric series best represents the repeating decimal [tex]$0.4444\ldots$[/tex].
To identify a repeating decimal, we can focus on the pattern of the digits. The given decimal [tex]$0.4444\ldots$[/tex] has the digit 4 repeating indefinitely.
We can express this repeating decimal in the form of a series as follows:
[tex]$
0.4444\ldots = 0.4 + 0.04 + 0.004 + 0.0004 + \ldots
$[/tex]
Each term in this series represents the decimal place value moved one position to the right, with each term being a fraction of 10. This can be written as:
[tex]$
0.4 = \frac{4}{10},\, 0.04 = \frac{4}{100},\, 0.004 = \frac{4}{1,000},\, \ldots
$[/tex]
Thus, the series can be expressed as:
[tex]$
\frac{4}{10} + \frac{4}{100} + \frac{4}{1,000} + \frac{4}{10,000} + \ldots
$[/tex]
Therefore, the geometric series representing [tex]$0.4444\ldots$[/tex] as a fraction is [tex]$\frac{4}{10} + \frac{4}{100} + \frac{4}{1,000} + \frac{4}{10,000} + \ldots$[/tex]
The correct answer is:
[tex]$
\frac{4}{10} + \frac{4}{100} + \frac{4}{1,000} + \frac{4}{10,000} + \ldots
$[/tex]
