To find the sum of the series
[tex]\[
1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots - \frac{1}{512},
\][/tex]
we recognize that this is a geometric series. For a geometric series, each term can be expressed as
[tex]\[
a, ar, ar^2, ar^3, \ldots,
\][/tex]
where [tex]\( a \)[/tex] is the first term, and [tex]\( r \)[/tex] is the common ratio.
In this particular series, the first term [tex]\( a \)[/tex] is 1, and the common ratio [tex]\( r \)[/tex] is -0.5. We need to determine how many terms there are in the series up to [tex]\(-\frac{1}{512}\)[/tex].
The term [tex]\(-\frac{1}{512}\)[/tex] occurs as the 10th term because:
- The sequence of denominators follows the form [tex]\(2^n\)[/tex] where [tex]\( n \)[/tex] starts from 0. Hence, [tex]\(\frac{1}{2^9}\)[/tex] corresponds to [tex]\(2^9 = 512\)[/tex].
- The sequence starts at [tex]\( n = 0 \)[/tex] and goes up to [tex]\( n = 9 \)[/tex], thus constituting 10 terms in total.
The formula to find the sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of a geometric series is:
[tex]\[
S_n = a \frac{1 - r^n}{1 - r}
\][/tex]
Substitute the known values [tex]\( a = 1 \)[/tex], [tex]\( r = -0.5 \)[/tex], and [tex]\( n = 10 \)[/tex]:
[tex]\[
S_{10} = 1 \cdot \frac{1 - (-0.5)^{10}}{1 - (-0.5)}
\][/tex]
Calculating [tex]\((-0.5)^{10}\)[/tex]:
[tex]\[
(-0.5)^{10} = 0.0009765625
\][/tex]
Next, we substitute [tex]\((-0.5)^{10}\)[/tex] into the sum formula:
[tex]\[
S_{10} = \frac{1 - 0.0009765625}{1 + 0.5}
\][/tex]
Simplify the equation:
[tex]\[
S_{10} = \frac{0.9990234375}{1.5}
\][/tex]
Finally, divide the numerator by the denominator:
[tex]\[
S_{10} = 0.666015625
\][/tex]
Therefore, the sum of the series is:
[tex]\[
\boxed{0.666015625}
\][/tex]
