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Evaluate the infinite geometric series
0.4 + 0.036 + 0.0000324 + ...
Express your answer as a fraction with integer numerator and denominator.

 Jun 16, 2024
 #1
avatar+1075 
+1

In order to solve this problem, we must find the common ratio of the numbers. 

Since it's a geometric series, setting r as the common ratio, we get

\(r = .036 / .4 = 36 / 400 = 9 /100 = .09 \)

 

We can calculate the sum using a really handy formula. 

\(\frac{a}{1-r}\) where a is the first number and r is the common ratio. 

 

Plugging in the values we already now, we get that the sum is equal to

\( .4 / [ 1 -.09] = .4 / .91 = 40 /91 \)

 

So our final answer is 40/91

 

Thanks! :)

 Jun 16, 2024

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