Thread: Can someone explain Geometric series?

This may be the wrong place to post this, but I'm going to ask here anyway...

For ages I've seen people using the sum ($\displaystyle \sum$) symbol with like n-1 on top and i=2 beneath it (or something along those lines), and I understand this is a geometric series, and I feel I almost understand it, but I thought if someone could try and explain it I wouldn't get confused ^_^

I'm probably going to ask my maths teacher anyway, but as it's a Sunday I thought I'd ask on here...

Any help much appreciated I missed the lesson on this in school so I don't fully understand


Oh and move this to a more relevant place if this is the wrong place to post

A geometric series is a sequence where the there is a constant ratio between each term.

eg, the geometric sequence where the multiplier is 1/2 and the first term is 3 would look like:

$\displaystyle 3, \frac{3}{2} , \frac{3}{4}, \frac{3}{8} , \frac{3}{16}, \frac{3}{16}, \frac{3}{32}, \frac{3}{64}, \frac{3}{128}...$




Suppose we wanted to find the sum of all terms in the series. We would do:

$\displaystyle 3 + \frac{3}{2} + \frac{3}{4}+ \frac{3}{8} + \frac{3}{16}, \frac{3}{16}+ \frac{3}{32}+ \frac{3}{64}+ \frac{3}{128}...$

unfortunately, you would never stop writing the sum down!

Instead, we notice that the series can be written:
$\displaystyle 3 + \frac{3}{2} + \frac{3}{2^2}+ \frac{3}{2^3} + \frac{3}{2^4}...$

$\displaystyle =\frac{3}{2^0} + \frac{3}{2^1} + \frac{3}{2^2}+ \frac{3}{2^3} + \frac{3}{2^4}...$

$\displaystyle =3\left(\frac{1}{2^0} + \frac{1}{2^1} + \frac{1}{2^2}+ \frac{1}{2^3} + \frac{1}{2^4}... \right)$

Which can be expressed as a single summation

$\displaystyle 3\sum_{i=0}^{\infty}\frac{1}{2^i}$

is that what you were asking? or did you want an explanation of why $\displaystyle 3\sum_{i=0}^{\infty}\frac{1}{2^i} = 3\left(\frac{1}{2^0} + \frac{1}{2^1} + \frac{1}{2^2}+ \frac{1}{2^3} + \frac{1}{2^4}... \right)$. in that case you might want to look at http://en.wikipedia.org/wiki/Summation

Thanks

That really helped, and yeah I was wondering why http://www.mathhelpforum.com/math-help/JSREPL85139:; ... But that example is simple and easy to understand so now I can like.... you know ^_^ anyway THANKS.