What's wrong with $\sum_{i=0}^{\infty}x^i = \frac{1}{1-x}$

Question

Set $$y=\sum_{i=0}^{\infty}x^i $$ Multiply both side by $x$, then we have $$yx=\sum_{i=0}^{\infty}x^{i+1}$$ Use the first one to minus the second one, we have $$y(1-x)=1$$ Then we have $$y=\frac{1}{1-x}$$, which means that

$$\sum_{i=0}^{\infty}x^i = \frac{1}{1-x}$$

But obviously only when $|{x}|<1$ the equation holds.

So what's wrong with the deduction process above? I opaquely heard about that it's related to the convergence domain, but I hardly get the hang of that.

Answer 1

The issues is (as others have mentioned) when you subtract the two series. The rules for adding or subtracting series say that you can only do this when you have convergent series. So if $\lvert x\lvert \geq 1$ then the series that you have set to be $y$ is not convergent. Hence you don't know that you can subtract the two series in that case.

The rule is that: If the series $\sum a_n$ is convergent with sum $a$ and $\sum b_n$ is convergent with sum $b$, then the series $\sum a_n + b_n$ is convergent with sum $a + b$.

Note again the requirement that you have to start with two series that you know are convergent.

But since you are trying to prove the formula, you don't a priori know when the series converges even for $\lvert x\lvert < 1$.

That is why you have you have to start with finite sums as is done in this Wikipedia article.

Answer 2

For $x\geq1$: $$y=\lim_{n\to\infty}\sum_{i=0}^nx^i=\infty,\\ yx=\lim_{n\to\infty}\sum_{i=0}^nx^{i+1}=\infty\Rightarrow\\ y-yx=\infty-\infty=?$$

Answer 3

Really, the problem is the way you operate on the first equality: when you let $y$ equal an infinite sum, $y$ could be possibly infinite, in which case, things like $y - yx$ is of the fishy form $\infty -\infty$.

Answer 4

When you subtract the two equations you might be subtracting infinity from infinity, which is not something you want to do.