Write out the sum. \[ \sum_{k=0}^{n-1} \frac{1}{5^{k+1}} \]

Step 1 ：Write out the sum: \(\sum_{k=0}^{n-1} \frac{1}{5^{k+1}}\)

Step 2 ：This is a geometric series with first term 1/5 and common ratio 1/5.

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

Step 4 ：In this case, a = 1/5, r = 1/5, and n = n (the number of terms is given by the upper limit of the summation minus the lower limit plus 1).

Step 5 ：We can plug these values into the formula to find the sum.

Step 6 ：The sum of the series is given by the formula \(\boxed{S = \frac{1}{5} \left(1 - \left(\frac{1}{5}\right)^n\right) \div \left(1 - \frac{1}{5}\right)}\)