Find the sum of the convergent series.

5 + 0.5 + 0.05 + 0.005 + . . .

The series given is:

\[
5 + 0.5 + 0.05 + 0.005 + \dots
\]

This is a geometric series where the first term \( a = 5 \) and the common ratio \( r = 0.1 \) (since each term is multiplied by 0.1 to get the next term).

The sum \( S \) of an infinite geometric series can be calculated using the formula:

\[
S = \frac{a}{1 - r}
\]

provided that \( |r| < 1 \). In this case, \( r = 0.1 \), which satisfies \( |r| < 1 \), so we can use the formula.

Substitute the values of \( a \) and \( r \):

\[
S = \frac{5}{1 - 0.1} = \frac{5}{0.9} = \frac{50}{9}
\]

Thus, the sum of the convergent series is:

\[
S = \frac{50}{9} \approx 5.56
\]

Would you like further details or clarification?

### Related Questions:
1. What is the general formula for the sum of a geometric series?
2. How do we determine if a geometric series converges or diverges?
3. Can a geometric series converge if \( |r| \geq 1 \)?
4. What are some real-life applications of converging geometric series?
5. How do you find the sum of a finite geometric series?

**Tip:** When solving geometric series problems, always check that the absolute value of the common ratio is less than 1 for convergence.

Learn how to find the sum of the infinite geometric series 5 + 0.5 + 0.05 + 0.005... with step-by-step calculations. Explore key concepts such as convergence, infinite series, and the geometric series formula. Suitable for students in Grades 9-12.

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