find the infinite sum of the series, if it exists.
2−0.4+0.08−0.016+0.0032...

To find the infinite sum of the given series, we first need to identify if it's a geometric series and, if so, find its common ratio.

The series given is:

\[
2 - 0.4 + 0.08 - 0.016 + 0.0032 \ldots
\]

Let's identify the first term (\(a\)) and the common ratio (\(r\)).

1. The first term \(a\) is:

\[
a = 2
\]

2. To find the common ratio \(r\), we divide the second term by the first term:

\[
r = \frac{-0.4}{2} = -0.2
\]

To verify, we can check if the ratio between consecutive terms remains the same:

\[
r = \frac{0.08}{-0.4} = -0.2
\]
\[
r = \frac{-0.016}{0.08} = -0.2
\]
\[
r = \frac{0.0032}{-0.016} = -0.2
\]

Since the ratio is consistent, this is indeed a geometric series with a common ratio \(r = -0.2\).

### Sum of an Infinite Geometric Series

The sum \(S\) of an infinite geometric series is given by:

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

provided that \(|r| < 1\). In our case, \(|-0.2| = 0.2\), which is less than 1, so the sum exists.

Now, plug in the values of \(a\) and \(r\):

\[
S = \frac{2}{1 - (-0.2)}
\]
\[
S = \frac{2}{1 + 0.2}
\]
\[
S = \frac{2}{1.2}
\]
\[
S = \frac{2}{\frac{6}{5}}
\]
\[
S = \frac{2 \times 5}{6}
\]
\[
S = \frac{10}{6}
\]
\[
S = \frac{5}{3}
\]

### Conclusion

The infinite sum of the series is:

\[
\boxed{\frac{5}{3}}
\]

Would you like more details or have any questions? Here are some related questions to further explore this topic:

1. How can you determine if a series is geometric or not?
2. What is the formula for the sum of the first \(n\) terms of a geometric series?
3. How do you calculate the sum of an infinite series if the common ratio is greater than 1?
4. Can the sum of an infinite geometric series be a negative number?
5. How would the sum change if the common ratio was \(-0.5\) instead of \(-0.2\)?

**Tip:** When dealing with infinite geometric series, always check the common ratio's absolute value. If \(|r| < 1\), the series converges and has a finite sum; otherwise, it diverges.

Explore how to find the infinite sum of the series 2, -0.4, 0.08, -0.016, 0.0032... This problem involves identifying it as a geometric series, finding the common ratio, and applying the formula for the sum of an infinite geometric series. Suitable for high school students.

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