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Consider two infinite geometric series. The first has leading term a, common ratio b, and sum S.The second has a leading term b, common ratio a,  and sum \(\frac{1}{S}\). Find the value of a+b.

 Feb 11, 2019
 #1
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+1

We have that

 

a / [ 1 - b]  =  S     (1)      and

 

b / [ 1 - a ] =  1/S        we can write this as

 

[ 1 - a ] / b =  S    (2)

 

Equating (1) and (2)  we have that

 

a / [ 1 - b ]  =  [ 1 - a ] /  b            cross-multiply

 

ab = [ 1 - b ] [ 1 - a ]         simplify

 

ab =  1 - a - b  + ab      subtract ab from both sides

 

0 = 1 - (a + b)           add (a + b) to both sides

 

(a + b )   =   1

 

 

cool cool  cool

 Feb 12, 2019

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