I need to find the value of this or show that it does not converge:

The summation from n=1 to infinity of $\displaystyle

\frac{{2^{n+1}+3^{n-1}}}{{5^n}}

$

Any help would be very appreciated.

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- Feb 4th 2008, 03:28 PMchaddyconverges
I need to find the value of this or show that it does not converge:

The summation from n=1 to infinity of $\displaystyle

\frac{{2^{n+1}+3^{n-1}}}{{5^n}}

$

Any help would be very appreciated. - Feb 4th 2008, 03:56 PMgalactus
Rewrite as:

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

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

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

Can you finish now?. - Feb 4th 2008, 05:07 PMchaddy
Okay, that helps a lot.

So after splitting it up like you did $\displaystyle

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 $\displaystyle \frac{4}{3}+\frac{1}{2}=\frac{11}{6}$

Is that correct? - Feb 4th 2008, 05:55 PMgalactus
Yep. That's right.