<PRE><B>Question 117354</B>
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---1.33x

Now what?&lt;pre&gt;&lt;font size = 4&gt;&lt;b&gt;  
----------------------------------------------------
Since the ratios are not equal, you have
shown this series is not a geometric series.

1 + 2x + 3x&lt;sup&gt;2&lt;/sup&gt; + 4x&lt;sup&gt;3&lt;/sup&gt; ...

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 + x&lt;sup&gt;2&lt;/sup&gt; + x&lt;sup&gt;3&lt;/sup&gt; + x&lt;sup&gt;4&lt;/sup&gt; + ททท + C

and this, all except for the arbitrary constant C IS 
a geometric series with a&lt;sub&gt;1&lt;/sub&gt; = x and r = x

Now we use the formula for the sum of an infinite
geometric series:

       a&lt;sub&gt;1&lt;/sub&gt;
S&lt;sub&gt;&lt;font face = &quot;symbol&quot;&gt;ฅ&lt;/font&gt;&lt;/sub&gt; = —————
      1-r

       x
S&lt;sub&gt;&lt;font face = &quot;symbol&quot;&gt;ฅ&lt;/font&gt;&lt;/sub&gt; = —————
      1-x
                               x
x + x&lt;sup&gt;2&lt;/sup&gt; + x&lt;sup&gt;3&lt;/sup&gt; + x&lt;sup&gt;4&lt;/sup&gt; + ททท + C = —————
                              1-x

So now we take the derivative term by term
and get:
                                (1-x)(1) - x(-1)
1 + 2x + 3x&lt;sup&gt;2&lt;/sup&gt; + 4x&lt;sup&gt;3&lt;/sup&gt; + ททท + 0 = ——————————————————
                                    (1-x)&lt;sup&gt;2&lt;/sup&gt;
   1 - x + x
= ———————————
     (1-x)&lt;sup&gt;2&lt;/sup&gt;  

     1
= ————————
   (1-x)&lt;sup&gt;2&lt;/sup&gt;

Edwin&lt;/pre&gt;</PRE>