Find the sum of the geometric series.
\[
11+\frac{11}{2}+\frac{11}{4}+\frac{11}{8}+\ldots
\]
The sum of the geometric series is (Type an integer or a simplified fraction.)

Answer

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Answer

Final Answer: The sum of the geometric series is \(\boxed{22}\)

Steps

Step 1 ：Given the geometric series \(11+\frac{11}{2}+\frac{11}{4}+\frac{11}{8}+\ldots\)

Step 2 ：We recognize that this is a geometric series with the first term \(a = 11\) and the common ratio \(r = \frac{1}{2}\)

Step 3 ：The sum of an infinite geometric series can be found using the formula \(S = \frac{a}{1 - r}\)

Step 4 ：Substituting the given values into the formula, we get \(S = \frac{11}{1 - \frac{1}{2}} = 22\)

Step 5 ：Final Answer: The sum of the geometric series is \(\boxed{22}\)