Find the infinite sum of the geometric sequence with $a=5, r=\frac{4}{5}$ if it exists.
\[
S_{\infty}=
\]
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Step 1 ：Given a geometric sequence with the first term \(a = 5\) and the common ratio \(r = \frac{4}{5}\).

Step 2 ：We are asked to find the infinite sum of this geometric sequence, denoted as \(S_{\infty}\).

Step 3 ：The formula for the sum of an infinite geometric sequence is \(S_{\infty} = \frac{a}{1 - r}\), provided that the absolute value of \(r\) is less than 1.

Step 4 ：Substitute \(a = 5\) and \(r = \frac{4}{5}\) into the formula, we get \(S_{\infty} = \frac{5}{1 - \frac{4}{5}}\).

Step 5 ：Solving the equation gives \(S_{\infty} = 25.000000000000007\).

Step 6 ：Rounding to the nearest whole number, we get \(S_{\infty} = 25\).

Step 7 ：Final Answer: The infinite sum of the geometric sequence is \(\boxed{25}\).