# Thread: Series Converge or diverge 2

1. ## Series Converge or diverge 2

consider the series
(1+1) + (0.3-.4642) + (0.03 + .2154) + (0.003 - .1000) + (.0003 + .0464) + ..........

I found the series to be:

$\displaystyle \Sigma_{n=0}^{\infty} {3(1/10)^n} + (-1)^{n+1}(1/\sqrt[3]{10})^n$

I found it to be convergent because $\displaystyle 3\Sigma_{n=0}^{\infty} (1/10)^n$converges by geometric series test because r= |1/10| < 1

and

$\displaystyle \Sigma_{n=0}^{\infty} (-1)^{n+1}(1/\sqrt[3]{10})^n$

converges by geometric series test becuse the |r| < 1 also

so...
the series

$\displaystyle \Sigma_{n=0}^{\infty} {3(1/10)^n} + (-1)^{n+1}(1/\sqrt[3]{10})^n$

would convergem but then I have to find the EXACT sum of the series and I don't know how to do this.

consider the series
(1+1) + (0.3-.4642) + (0.03 + .2154) + (0.003 - .1000) + (.0003 + .0464) + ..........

I found the series to be:

$\displaystyle \Sigma_{n=0}^{\infty} {3(1/10)^n} + (-1)^{n+1}(1/\sqrt[3]{10})^n$

I found it to be convergent because 3\Sigma_{n=0}^{\infty} (1/10)^n converges by geometric series test because r= |1/10| < 1

and

$\displaystyle \Sigma_{n=0}^{\infty} (-1)^{n+1} (1/\sqrt[3]{10})^n$
converges by geometric series test becuse the |r| < 1 also

so...
the series
math]
\Sigma_{n=0}^{\infty} {3(1/10)^n} + (-1)^{n+1}(1/\sqrt[3]{10})^n
[/tex]

would convergem but then I have to find the EXACT sum of the series and I don't know how to do this.
$\displaystyle \sum_{n=0}^{\infty}\left\{3\left(\frac{1}{10}\righ t)^n-\left(\frac{-1}{\sqrt[3]{10}}\right)^n\right\}$

You chould know that $\displaystyle \forall{x}\backepsilon|x|<1~\sum_{n=0}^{\infty}x^n =\frac{1}{1-x}$

So your sum is $\displaystyle \frac{3}{1-\frac{1}{10}}-\frac{1}{1+\frac{1}{\sqrt[3]{10}}}$