Question

Calculate the sum of the infinite series S subscript infinity baseline equals 20 minus 10 plus 5 minus ellipsis.
(1 point)
Responses

Start Fraction 40 over 3 End Fraction
Image with alt text: Start Fraction 40 over 3 End Fraction

Start Fraction 20 over 3 End Fraction
Image with alt text: Start Fraction 20 over 3 End Fraction

40
40

It does not converge.

Answer 1

To find the sum of the infinite series \( S = 20 - 10 + 5 - \ldots \), we begin by establishing the pattern of the series. It can be rewritten as:

\[
S = 20 - 10 + 5 - 2.5 + \ldots
\]

We can see that this is a geometric series where the first term \( a = 20 \) and the common ratio \( r = -\frac{1}{2} \) (since each term is multiplied by \(-\frac{1}{2}\) to get the next term).

The sum of an infinite geometric series can be calculated using the formula:

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

As long as \( |r| < 1 \).

Here, since \( |r| = \frac{1}{2} < 1 \), we can apply the formula:

1. First, determine \( a \) and \( r \):
- \( a = 20 \)
- \( r = -\frac{1}{2} \)

2. Substitute \( a \) and \( r \) into the formula:
\[
S = \frac{20}{1 - \left(-\frac{1}{2}\right)}
\]

Calculating the denominator:
\[
1 - \left(-\frac{1}{2}\right) = 1 + \frac{1}{2} = \frac{3}{2}
\]

Putting it all together:
\[
S = \frac{20}{\frac{3}{2}} = 20 \cdot \frac{2}{3} = \frac{40}{3}
\]

Thus, the sum of the infinite series is \( \frac{40}{3} \).

So, the correct answer is:

\(\frac{40}{3}\)