Question

Evaluate the infinite sum sigma-summation Underscript i equals 1 Overscript infinity EndScripts Start Fraction 4 over 7 End Fraction left parenthesis Start Fraction 7 over 6 End Fraction right parenthesis superscript i minus 1 baseline.
(1 point)
Responses

Start Fraction 49 over 18 End Fraction
Image with alt text: Start Fraction 49 over 18 End Fraction

Start Fraction twenty-four over seven End Fraction
Image with alt text: Start Fraction twenty-four over seven End Fraction

negative Start Fraction 24 over 7 End Fraction
Image with alt text: negative Start Fraction 24 over 7 End Fraction

It does not converge.

Answer 1

To evaluate the infinite sum

\[
\sum_{i=1}^{\infty} \frac{4}{7} \left( \frac{7}{6} \right)^{i-1},
\]

we can recognize that this is a geometric series.

The general form for the sum of an infinite geometric series \(\sum_{n=0}^{\infty} ar^n\) is given by:

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

where \(a\) is the first term of the series, and \(r\) is the common ratio, provided that \(|r| < 1\).

In our case, rewriting the sum starting from \(i = 1\) to \(i = \infty\), we can note that:

- \(a = \frac{4}{7}\) (this is the factor outside the summation).
- The factor \(\left( \frac{7}{6} \right)^{i-1}\) indicates that \(r = \frac{7}{6}\).

However, since the series starts from \(i=1\), we can express it as:

\[
\sum_{i=1}^{\infty} ar^{i-1} = a \sum_{n=0}^{\infty} r^{n}.
\]

Here, by recognizing \(a = \frac{4}{7}\) and \(r = \frac{7}{6}\), we can calculate the sum. Notice the issue with convergence:

We need to check the convergence criterion which states that the infinite geometric series converges only if \(|r| < 1\). In this case,

\[
| \frac{7}{6} | = \frac{7}{6} > 1.
\]

Since \(|r| > 1\), the series does not converge.

Thus, the answer is:

**It does not converge.**