SOLUTION: {{{a=1+2x+4x^2+..............}}}, where -1<2x<1, {{{b= 1+3y+9y^2+.............}}}, where -1<3y<1 and 3y+2x=1, prove that ab=a+b

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Question 622156: , where -1<2x<1, , where -1<3y<1 and 3y+2x=1, prove that ab=a+b
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
a =  = 

That's an infinite geometric series with first term t1 = 1,
and r = 2x.  And since -1 < 2x < 1 it converges and we can use the formula:

 =  = 


b =  = 


That's also an infinite geometric series with first term t1 = 3y,
and r = 3y.  And since -1 < 3y < 1 we can use the formula:

 =  =  

So 

a = , b = 

ab =  · = 

a + b = +{1/(1-3y)}}} =  =

 =  =



and we are given that 3y+2x=1, so we replace(3y+2x) by 1

 = 

So ab and a + b  both equal to 

Edwin

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