# converges

• Feb 4th 2008, 03:28 PM
converges
I need to find the value of this or show that it does not converge:
The summation from n=1 to infinity of $
\frac{{2^{n+1}+3^{n-1}}}{{5^n}}
$

Any help would be very appreciated.
• Feb 4th 2008, 03:56 PM
galactus
Rewrite as:

$2\sum_{n=1}^{\infty}(\frac{2}{5})^{n}+\frac{1}{3}\ sum_{n=1}^{\infty}(\frac{3}{5})^{n}$

Now, the sums are easier. It's just a matter of a geometric series.

For instance, $\sum_{n=1}^{\infty}(\frac{2}{5})^{n}=\frac{\frac{2 }{5}}{1-\frac{2}{5}}=\frac{2}{3}$

Can you finish now?.
• Feb 4th 2008, 05:07 PM
Okay, that helps a lot.
So after splitting it up like you did $
2\sum_{n=1}^{\infty}(\frac{2}{5})^{n}+\frac{1}{3}\ sum_{n=1}^{\infty}(\frac{3}{5})^{n}
$

I can rewrite this as $\frac{4}{3}+\frac{1}{2}=\frac{11}{6}$
Is that correct?
• Feb 4th 2008, 05:55 PM
galactus
Yep. That's right.