SOLUTION: Find the sum of the series 1 + 2x + 3x^2 + 4x^3 ... I cannot understand how to find the ratio Like 1 + 2x + 3x^2 + 4x^3 ... --\/---\/-------\/-------- --2x---1.5x---1.33

  
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Since the ratios are not equal, you have
shown this series is not a geometric series.

1 + 2x + 3x2 + 4x3 ...

I hope you are taking calculus, for ordinary
algebra cannot handle this sort of problem.
 
If we take the antiderivative term by term,
we get

x + x2 + x3 + x4 + ··· + C

and this, all except for the arbitrary constant C IS 
a geometric series with a1 = x and r = x

Now we use the formula for the sum of an infinite
geometric series:

       a1
S¥ = —————
      1-r

       x
S¥ = —————
      1-x
                               x
x + x2 + x3 + x4 + ··· + C = —————
                              1-x

So now we take the derivative term by term
and get:
                                (1-x)(1) - x(-1)
1 + 2x + 3x2 + 4x3 + ··· + 0 = ——————————————————
                                    (1-x)2
   1 - x + x
= ———————————
     (1-x)2  

     1
= ————————
   (1-x)2

Edwin