Evaluate the infinite geometric series
\[0.4 + 0.08 + 0.016 + 0.0032 + \dotsb.\]Express your answer as a fraction with integer numerator and denominator.


The given series is an infinite geometric series:

\[
0.4 + 0.08 + 0.016 + 0.0032 + \dotsb
\]

In a geometric series, the general form is:

\[
S = a + ar + ar^2 + ar^3 + \dotsb
\]

where \(a\) is the first term and \(r\) is the common ratio between consecutive terms.

### Step 1: Identify the first term \(a\) and the common ratio \(r\)
- The first term \(a = 0.4\).
- To find the common ratio \(r\), divide the second term by the first term:
\[
r = \frac{0.08}{0.4} = 0.2
\]

Thus, the common ratio \(r = 0.2\).

### Step 2: Apply the sum formula for an infinite geometric series
The sum \(S\) of an infinite geometric series is given by the formula:

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

where \( |r| < 1 \).

### Step 3: Plug in the values of \(a\) and \(r\)

Substitute \(a = 0.4\) and \(r = 0.2\) into the formula:

\[
S = \frac{0.4}{1 - 0.2} = \frac{0.4}{0.8}
\]

### Step 4: Simplify the expression
Simplifying the fraction:

\[
S = \frac{0.4}{0.8} = \frac{4}{8} = \frac{1}{2}
\]

Thus, the sum of the infinite geometric series is:

\[
S = \frac{1}{2}
\]

### Final Answer:
\[
\boxed{\frac{1}{2}}
\]

Would you like to explore further details or have any questions?

Here are 5 related questions to expand on this:

1. What happens if the common ratio \(r\) is greater than 1 in an infinite geometric series?
2. Can you evaluate the sum of a finite geometric series instead of an infinite one?
3. How would you apply the geometric series sum formula if the first term were negative?
4. What is the convergence condition for an infinite geometric series?
5. How can the geometric series concept be applied to real-world problems?

**Tip:** In an infinite geometric series, the sum converges only if the common ratio \( |r| < 1 \).

Evaluate the infinite geometric series 0.4 + 0.08 + 0.016 + 0.0032 + ... Express your answer as a fraction with integer numerator and denominator.

Learn how to evaluate the infinite geometric series 0.4 + 0.08 + 0.016 + 0.0032 + ... using the formula for an infinite geometric series. This problem covers key concepts such as identifying the first term, the common ratio, and applying the geometric series sum formula. Suitable for Grades 10-12.

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