Problem 16. (7 points) Determine the sum of the following series. \[ \sum_{n=1}^{\infty}\left(\frac{1^{n}+9^{n}}{11^{n}}\right) \]

Step 1 ：Define the first term and common ratio for both series. For the first series, the first term \(a_1\) is \(\frac{1}{11}\) and the common ratio \(r_1\) is \(\frac{1}{11}\). For the second series, the first term \(a_2\) is \(\frac{9}{11}\) and the common ratio \(r_2\) is \(\frac{9}{11}\).

Step 2 ：Calculate the sum of both series. The sum of a geometric series is given by the formula \(\frac{a}{1 - r}\), where \(a\) is the first term and \(r\) is the common ratio. Therefore, the sum \(S_1\) of the first series is \(\frac{a_1}{1 - r_1} = 0.1\) and the sum \(S_2\) of the second series is \(\frac{a_2}{1 - r_2} = 4.500000000000002\).

Step 3 ：Add the sums together to get the sum of the original series. Therefore, the sum \(S\) of the original series is \(S_1 + S_2 = 4.600000000000001\).

Step 4 ：Simplify the final answer to get \(\boxed{4.6}\).