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Looking at the following power series,

$$\sum _{n=1}^{\infty} x^n = 1+x+x^2...$$

I am trying to find the values of $x$ that make the series converge, and what function it converges to.

My thinking was to use the ratio test,

$$\lim_{n \to \infty}|\frac{a_{n+1}}{a_n}| = \lim_{n\to\infty}|\frac {x^{n+1}}{x^n}|= x$$

For $|x|<1$ the series converges by the ratio test.
For $|x|>1$ the series diverges by the tatio test.

But what about $|x|=1$, how do I check if this converges or not? By the ratio test this in inconclusive

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    OBS: Using the ratio test to assert convergence/divergence of the geometric series is shooting your own bullets. – 2017-02-17
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    Note that whichever test it is, that gives a radius of convergence, it'll never specify what to do when |x|=R. You have to study the behaviour specifically. – 2017-02-17
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    @AloizioMacedo OBS? Not sure I've seen that acronym. I agree though. – 2017-02-17

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When $|x|=1$, your sum is either $$\sum_{n=1}^{\infty}1^n\quad\text{or}\quad \sum_{n=1}^{\infty}(-1)^n.$$ In both cases, $\displaystyle\lim_{n\to\infty}a_n\neq 0$ so they diverge.