The integral test is applicable if f is positive, continuous and decreasing function on infinite interval `[k,oo)` where `k>=1` and `a_n=f(x)` . Then the series `sum_(n=1)^ooa_n` converges or diverges if and only if the improper integral `int_1^oof(x)dx` converges or diverges.
For the given series `a_n=(4n)/(2n+1)`
Refer to the attached graph of the function. From the graph we observe that the function is positive and continuous. However it is not decreasing on the interval `[1,oo)`
We can also determine whether the function is decreasing by finding the derivative f'(x) such that `f'(x)<0` for `x>=1`
Let's find the derivative by the quotient rule:
which implies that the function is not decreasing.
Since the function does not satisfies the conditions for the integral test, we can not apply integral test.