1. ## Geometric Series

I was having trouble with this question and wondered if I could get some help with it ?

How many terms of the geometric series $\displaystyle 1 + \frac{2}{3} + \frac{4}{9}$ + ... must be taken so that the sum differs from 3 by less than $\displaystyle 10^{-3}$?

I started with this inequality:

$\displaystyle 3 - 10^{-3} < \frac{(\frac{2}{3})^n - 1}{\frac{2}{3} - 1} < 3 + 10^{-3}$

and ended up with this:

$\displaystyle \frac{ln(\frac{-1}{3000})}{ln(\frac{2}{3})} < n < \frac{ln(\frac{1}{3000})}{ln(\frac{2}{3})} = 19.74$

Of course you cannot take the log of a negative number so how do I work out what n must be greater than? Also sorry I have no idea how to do that maths code thing so it might look a bit messy, Thanks

2. ## codes

here is the page the explains codes.

3 - 10^(-3 < (2/3)^n - 1 )/(2/3 - 1) < 3 + 10 ^-3 would be

$\displaystyle 3 - 10^ < \frac{(\frac{2}{3})^n - 1 )}{(\frac{2}{3} - 1) < 3 + 10^ (-3}$

u have to play with it.

3. Hello, slevvio!

I discovered that there is a strange step in this solution . . .

How many terms of the geometric series $\displaystyle 1 + \frac{2}{3} + \frac{4}{9} + \cdots$
must be taken so that the sum differs from 3 by less than $\displaystyle 10^{-3}$ ?

The sum of the first n terms is: .$\displaystyle S_n \;=\;\frac{1 - (\frac{2}{3})^n}{1-\frac{2}{3}} \;=\;3\left[1-\left(\frac{2}{3}\right)^n\right]$

We want: .$\displaystyle \left|\,3\left[1-\left(\frac{2}{3}\right)^n\right] - 3\,\right| \;<\;0.001$

Then: .$\displaystyle -0.001 \;<\;3\left[1 - \left(\frac{2}{3}\right)^n\right] - 3 \;<\;0.001$

Divide by 3: .$\displaystyle -\frac{0.001}{3} \;<\;1 - \left(\frac{2}{3}\right)^n - 1 \;<\;\frac{0.001}{3}\quad\Rightarrow\quad -\frac{0.001}{3} \;<\:\underbrace{-\left(\frac{2}{3}\right)^n \;<\;\frac{0.001}{3}}$

The right half of the inequality can be discarded.
. . (Of course, a negative fraction is less than a positive fraction.)

So we have: .$\displaystyle -\frac{0.001}{3} \;<\;-\left(\frac{2}{3}\right)^n \quad\Rightarrow\quad -\left(\frac{2}{3}\right)^n \;>\;-\frac{0.001}{3}$

Multiply by -1: .$\displaystyle \left(\frac{2}{3}\right)^n \;< \;\frac{0.001}{3}$

Take logs: .$\displaystyle \ln\left(\frac{2}{3}\right)^n \;<\;\ln\left(\frac{0.001}{3}\right)\quad\Rightarr ow\quad n\cdot\ln\left(\frac{2}{3}\right) \;<\;\ln\left(\frac{0.001}{3}\right)$

Divide both sides by $\displaystyle \ln\left(\frac{2}{3}\right)$ . . . note that this quantity is negative.

. . $\displaystyle n \;> \;\frac{\ln\left(\frac{0.001}{3}\right)}{\ln\left( \frac{2}{3}\right)} \;= \;19.7461...$

Therefore: .$\displaystyle n \:\geq\:20$

4. Thank you you certainly are a super member!