Confirm that the integral test can be applied to the series. then use the integral test to determine the convergence or divergence of the series. 1 5 + 1 7 + 1 9 + 1 11 + 1 13

The question is:

Confirm that the integral test can be applied to the series. Then use the integral test to determine the convergence or divergence of the series: 1/5 + 1/7 + 1/9 + 1/11 + 1/13 + ...

The series is divergent.

Step-by-step explanation:

Given the series:

1/5 + 1/7 + 1/9 + 1/11 + 1/13 + ...

We can rewrite this series in summation notation as

Sum from 1 to infinity [1/(2n + 3)]

For this test to work,

1/(2n + 3) must be positive, decreasing, and continuous.

The summation is convergent if the integral from 1 to infinity [1/(2n + 3)] dx is convergent, and divergent if the integral from 1 to infinity [1/(2n + 3)] dx is divergent.

Now, let us consider the integral from 1 to infinity [1/(2x + 3)] dx

This can be written as

limit as y -> infty{integral from 1 to y [1/(2x + 3)] dx}

= (1/2)limit as y -> infty{ln|2x + 3| from 1 to y}

= (1/2)limit as y -> infty{ln|2y + 3| - ln(5)}

= infinity

Therefore, the series diverges.