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.
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?.
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?