Calculus of a Single Variable, Chapter 9, 9.3, Section 9.3, Problem 19

sum_(n=1)^oo(4n)/(2n+1)
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)
Consider f(x)=(4x)/(2x+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:
f(x)=(4x)/(2x+1)
f'(x)=((2x+1)d/dx(4x)-(4x)d/dx(2x+1))/(2x+1)^2
f'(x)=((2x+1)(4)-(4x)(2))/(2x+1)^2
f'(x)=(8x+4-8x)/(2x+1)^2
f'(x)=4/(2x+1)^2
So f'(x)>0
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.