Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$ Why does the following hold:
\begin{equation*}
\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ?
\end{equation*}
Can we generalize the above to 

$\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ?

Are there some values of $x$ for which the above formula is invalid?
What about if we take only a finite number of terms? Is there a simpler formula?

$\displaystyle \sum_{n=0}^{N} x^n$

Is there a name for such a sequence?

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If u expand your summation you get series $$1+0.7+0.7^2+\dots$$ as it is geometric series (you can see it here http://en.wikipedia.org/wiki/Geometric_series) $$\sum_{n=0}^{\infty}{0.7^n}=\frac{1}{1-0.7}$$
Or 
$S=1+x+x^2+\dots$
$xS_n=x+x^2+\dots=S-1$
now u take
$S-xS=1$
$S(1-x)=1
\implies S=\frac{1}{1-x}$
here your $x=0.7$
