Question

Evaluate the infinite geometric series 0.79 + 0.079 + 0.0079 + 0.00079 + 0.000079. Express your answer as a fraction with integer numerator and denominator.

Answer 1

Answer:

The fraction is 79/99

Step-by-step explanation:

We can express the given infinite geometric series as follows:

$\frac{79}{10^{2} }+\frac{79}{10^{3} }+\frac{79}{79^{4} }+\frac{79}{10^{5} }+...$

Which is an infinite geometric series with $r=\frac{1}{10^{2} }<1$ that converges, then its sum is:

$S=\frac{a}{1-r}$

Where a is the first factor of the serie: $a=\frac{79}{10^{2} }$

Replacing values:

$S=\frac{\frac{79}{10^{2} } }{1-\frac{1}{10^{2} } }  \\S=\frac{\frac{79}{10^{2} } }{\frac{99}{10^{2} } }\\S=\frac{79}{99}$