How do I find the sum of a geometric series if it doesn't seem consistent?

Question

I'm supposed to find the sum of the geometric series $$S= 9 + x + x^2 + x^3 + ...$$ I have no idea how to do this, since if I'm to follow the general pattern of infinite geometric series, it should be $$a + ax + ax^2 +......$$and so on, but the 9 doesn't fit since if the $x$ is $x/9$, the third term would be $x^2/9$ and so on. What do I do for this?

Answer 1

This is simpler than you're thinking it is. The form you desire to pop up (at least in order to make some nice simplifications) is

$$1+x+x^2+x^3+\cdots$$

right? What would happen if you added $8$ to it? You would get

$$8+(1+x+x^2+x^3+\cdots)$$

or, equivalently,

$$9+x+x^2+x^3+\cdots$$

That is, these last two expressions are equivalent! The middle expression, provided $|x|<1$, lets us convert the infinite summation to a ratio in the usual way as well:

$$8+(1+x+x^2+x^3+\cdots) = 8 + \frac{1}{1-x}$$