$\sum_{n=0}^{\infty} \frac{8+\sin n}{11^{n}}$

Step 1 ：We are given the infinite series \(\sum_{n=0}^{\infty} \frac{8+\sin n}{11^{n}}\).

Step 2 ：This is an infinite geometric series with the first term a = \(\frac{8+\sin 0}{11^{0}}\) = 8 and the common ratio r = \(\frac{1}{11}\).

Step 3 ：The sum of an infinite geometric series can be calculated using the formula \(\frac{a}{1-r}\), where a is the first term and r is the common ratio.

Step 4 ：However, the \(\sin n\) term complicates things a bit. Since \(\sin n\) oscillates between -1 and 1, the series does not converge to a single value. Instead, it oscillates between two values.

Step 5 ：We can calculate these two values by considering the cases where \(\sin n\) is -1 and 1 separately.

Step 6 ：For \(\sin n = -1\), a1 = 8.0 and for \(\sin n = 1\), a2 = 8.841470984807897. The common ratio r = 0.09090909090909091.

Step 7 ：Using the formula for the sum of an infinite geometric series, we find that when \(\sin n = -1\), the sum is 8.8 and when \(\sin n = 1\), the sum is 9.725618083288687.

Step 8 ：The sum of the series oscillates between the two values we calculated, 8.8 and 9.73. This is because the \(\sin n\) term causes the terms of the series to oscillate between two values.

Step 9 ：Therefore, the series does not converge to a single value, but oscillates between these two values.

Step 10 ：Final Answer: The sum of the series oscillates between \(\boxed{8.8}\) and \(\boxed{9.73}\).