Respuesta :
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 + ...
Answer:
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.
