<s>[INST] 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?
 [/INST] Note that the sum $\sum_{n=0}^{\infty}r^{n}=\frac{1}{1-r}$ for $|r|<1$. According to this fact $\sum_{n=0}^{\infty}\frac{2^{n}+3^{n}}{6^{n}}=\sum_{n=0}^{\infty}\frac{2^{n}}{6^{n}}+\sum_{n=0}^{\infty}\frac{3^{n}}{6^{n}}=\sum_{n=0}^{\infty}(\frac{2}{6})^{n}+\sum_{n=0}^{\infty}(\frac{3}{6})^{n}=\frac{1}{1-1/3}+\frac{1}{1-1/2}=\frac{7}{2}$.
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