Sum formula for geometric series

Question

I solved this equation something like this, as shown in the photo:

Is it correct?

If I put $x=2$ I get weird results!

Answer 1

As a lot of the comments and the other answers point out, this only works for $-1<x<1$. I want to go in a slightly different direction: If you want to make sense of the function $\frac{x}{1-x}$ as a geometric series outside of that interval, we have: $$ -1 - \frac 1x - \frac{1}{x^2} - \frac{1}{x^3} - \cdots $$ which is a sequence that converges as long as $|x|> 1$, and it gives $\frac{x}{1-x}$ (or really, $\frac{1}{1/x-1}$, which ammounts to the same thing) if you do the same trick as you've done in your question. This is what is called "the series expansion of $\frac x{1-x}$ around $\infty$" (since the series converges as long as $x$ is large enough), while $x + x^2 + x^3 + \cdots$ is the series expansion around $0$ (since it converges as long as $x$ is close enough to $0$ [there is a bit more to it than that, but that's details]). Together, they give you geometric series that evaluate to $\frac{x}{x-1}$ on the whole number line except at $-1$ and $1$.

Answer 2

It's true that

$$x + x^2 + x^3 + ... = \frac{x}{1-x} $$

provided that $|x|< 1$. Therefore you cannot apply it when $x=2$.

In short, the reason it does not work when $|x|>1$ is because the terms blow up (the series is said to be divergent).

Answer 3

You assumed in your proof that the quantity $x + x^2 + x^3 + \cdots$ is a number. Indeed, whenever $(x + x^2 + x^3 + \cdots)$ is a quantity satisfying the formal rules of arithmetic, we must have $$ x + x^2 + x^3 + \cdots = \frac{x}{1-x} $$ as you have proven. However, under the usual considerations, $x + x^2 + x^3 + \cdots$ is not a real number. For example, if $x > 1$, it is more typical to say something like $$ x + x^2 + x^3 + \cdots = \infty $$ and $\infty$ does not satisfy the formal rules of arithmetic (for example, $\infty - 1 = \infty$).

It turns out that your formula only makes sense in our usual understanding of infinite sums when $|x| < 1$. Perhaps you can see how $x + x^2 + x^3 + \cdots$ should only make sense if the terms $x,x^2,x^3,\dots$ get smaller and smaller.