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I want to know how many values the $x$ takes so that the sum total $(0.5) ^ x * 0.5$ equals $1$.

But since the ratio is $0.5$, each term is less than the previous one, and the progression is approaching $0$ as the number of its terms increases.

That means that at some point the sum gives 1, but I will never know at what $ x $ value will that happen?

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    You mean the equation $0.5\cdot \sum_{x=0}^C 0.5^x=1$ ? And you want to find $C$ ? – 2017-02-06
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    Exactly, that's what I mean. – 2017-02-06
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    Ok. I think I get it. – 2017-02-06
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    You can ask a new question. – 2017-02-06

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The closed formula for the partial sum $$S_n=\sum_{x=0}^{n-1} r^x$$ is

$S_n=r\cdot \frac{1-r^n}{1-r}$

Thus $$0.5\cdot \sum_{x=0}^{n-1} 0.5^x=0.5\cdot \frac{1-0.5^n}{1-0.5}$$

$=1-0.5^n$

For what value of $n$ is the expression above equal to one ?

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    None? So the sum tends to 1 but it is never 1? – 2017-02-06
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    @CarlosFrostte That´s it. $\checkmark$ – 2017-02-06
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    I was asked to graph the probability functions of discrete random variables that are distributed geometrically. What I do? – 2017-02-06
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    The pdf of a geometric function is $f_X(x)=p\cdot (1-p)^{x-1}$. For $p=0.5$ it is $f_X(x)=0.5\cdot (0.5)^{x-1}$. The first values are $x=1,y=0.5; x=2,y=0.5^2; x=3,y=0.5^3,...$ I think you can graph this points. – 2017-02-06
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    I did, but I thought maybe I could do something else. Thank you. – 2017-02-06
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    You are welcome. One note: $n\to \infty 1-0.5^n=1$. So it $is \ equal$ to $1$ – 2017-02-06
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You can use formula -

$S_n = \frac{a(r^n - 1)}{r - 1}$

Where $S_n = 1$ and a is first term.