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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.

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4 Answers 4

up vote 8 down vote accepted

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.

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1  
A small TeXnical tip: if you use \lvert, then you should also use \rvert. Thus, to denote the absolute value of x, the best usage is \lvert x \rvert. Note that |x| also works (but |\sin x| has spacing issues that \lvert \sin x \rvert does not have). –  Hendrik Vogt Jan 12 '13 at 15:56
    
@HendrikVogt: Thanks for the tip! It actually makes sense now that you say it with the l in lvert being left. I will try to do that from now on. –  Thomas Jan 12 '13 at 15:57

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

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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 $\infty -\infty$.

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

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I give the first answer and I get the least number of votes up. SMH –  mathemagician Jan 12 '13 at 15:15
    
Answers are voted on not just by when they arrive, but also corresponding to their quality. –  Peter Olson Jan 13 '13 at 0:19

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