Finding the sum of series $\sum_{n=0}^∞ \frac{2^n + 3^n}{6^n}$ I am being asked to find the sum of the following convergent series :
$$\sum_{n=0}^∞ \frac{2^n}{6^n} + \frac{3^n}{6^n}$$
Attempting to generalize from partial sums yields nothing of interest:
$s_1 = \frac{5}{6}$
$s_2 = \frac{5}{6} + \frac{13}{36} = \frac{43}{36}$
$s_3 = \frac{43}{36} + \frac{35}{216} = \frac{293}{216}$
$s_4 = \frac{293}{216} + \frac{97}{1296} = \frac{1855}{1296} $
I do not see a pattern here...
How must I proceed to find the sum of this series?

$\frac{2^n}{6^n}=(\frac{1}{3})^n$ and $\frac{3^n}{6^n}=(\frac{1}{2})^n$
Now, use the well known result that for $|x|<1$,
$$\sum \limits_{n=0}^{\infty}x^n=\frac{1}{1-x}$$
