the limiting sum of 3 + x + x^2 + ..... is 18. If |x|<1, find x
how do you do this??
Everything after the 3 is an infinite geometric series.
For any infinite geometric series of the form
$\displaystyle S_n = a + ar + ar^2 + ar^3 + \dots$
converges to $\displaystyle \frac{a}{1 - r}$, provided $\displaystyle |r| < 1$.
In your case, the infinite geometric series is
$\displaystyle S_n = x + x^2 + x^3 + x^4 + \dots$.
Can you see what $\displaystyle a$ and $\displaystyle r$ have to be?
Can you see what it converges to if $\displaystyle |x| < 1$?
No.
I said the terms AFTER the 3.
In other words, your equation is
$\displaystyle 3 + \left(x + x^2 + x^3 + \dots\right) = 18$.
We are only dealing with the stuff inside the brackets.
Edit: It might help if you subtract the 3 to the other side.
So the equation becomes
$\displaystyle x + x^2 + x^3 + \dots = 15$.