Thread: Can someone explain Geometric series?

1. 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

2. 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

3. 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.