<s>[INST] How to evaluate series. eg. $a_n = \frac{1}{2^n}$ 
Possible Duplicate:
Value of $\sum\limits_n x^n$ 

In my lecture notes: 
$$a_n = \frac{1}{2^n}, \qquad \sum_{n-1}^{\infty} a_n = \lim_{n\to\infty} (1-\frac{1}{2^n}) = 1$$
How do I get $(1-\frac{1}{2^n})$? 
A similar example given is
$$a_n = 2^{n-1}, \qquad \sum_{n-1}^{\infty} a_n = \lim_{n\to\infty} (2^n - 1)$$
How do I get these? 
 [/INST] $$a_n = \frac{1}{2^n}, \qquad \sum_{n=1}^{\infty} a_n = \lim_{n\to\infty} (1-\frac{1}{2^n}) = 1$$
Depends how you interpret it.  First let us look at the solution to
First we will be using (for  $r<1$)
$$
\sum_{k=0}^{n} ar^{k} = \frac{a- ar^{n+1}}{1-r}
$$
$$
\sum_{k=0}^\infty ar^k = \frac{a}{1-r}
$$
Now in the problem above
$$a_n = \frac{1}{2^n}$$  and therefore 
$$
\begin{align*}
\sum_{n=1}^{\infty} a_n &= \sum_{n=1}^{\infty}\frac{1}{2^n}\\
&= {\frac{1}{2}} ( 1 + \frac{1}{2^{1}} + \frac{1}{2^{2}} + \dots )\\
&= {\frac{1}{2}} (\frac{1}{1-\frac{1}{2}})
&= 1
\end{align*}
$$
And 
$$\lim_{n\to\infty} (1-\frac{1}{2^n}) = 1 - \lim_{n\to\infty} \frac{1}{2^n} = 1$$
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