Thread: limiting sum of a series??

1. limiting sum of a series??

the limiting sum of 3 + x + x^2 + ..... is 18. If |x|<1, find x
how do you do this??

2. Originally Posted by deej813
the limiting sum of 3 + x + x^2 + ..... is 18. If |x|<1, find x
how do you do this??
$\displaystyle 3+x+x^2+...=18$

$\displaystyle 3+\frac{x}{1-x}=18$

$\displaystyle \frac{x}{1-x}=15$

$\displaystyle x=15-15x$

$\displaystyle 16x=15$

$\displaystyle x=\frac{15}{16}$

3. how does that work??
where does the x/1-x come from??

4. 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 $\displaystyle |x| < 1$?

5. wouldn't it be 3/1-x cos isn't a=3??

6. 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

$\displaystyle x + x^2 + x^3 + \dots = 15$.

7. oh ok i get it
thanks