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Evaluate the infinite geometric series: \(1-\frac{2}{7}+\frac{4}{49}-\frac{8}{343}+\dots\)

 Jun 29, 2020
 #1
avatar+307 
+2

Leave the 1 out for now. Combine -2/7 and 4/49. This equals -10/49. Combine -4/49 and 16/2401. This equals -40/2401. 

This forms an infinite geometric series with first term -10/49 and common ratio 4/49. (You can go on a few more times to see that it is in fact a geometric series).

The sum is \(\frac{-10/49}{1-\frac{4}{49}}=\frac{-2}{9}\). We add this to our beginning 1 to find our final answer 1-2/9=7/9

 Jun 29, 2020
edited by thelizzybeth  Jun 29, 2020
 #4
avatar+794 
-1

Thank you!!

AnimalMaster  Jun 30, 2020
 #2
avatar+31340 
+1

The common ratio is simply r = -2/7, so the infinite sum is given by 1/(1 - (-2/7))  or 7/9

 Jun 30, 2020
 #3
avatar+794 
-1

Thank you!!

AnimalMaster  Jun 30, 2020

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