Answer
Function satisfies conditions for integral test
Series diverges
Work Step by Step
$\dfrac {1}{3}+\dfrac {1}{5}+\dfrac {1}{7}+\ldots =\sum ^{\infty }_{n=1}\dfrac {1}{2n+1}$
In order for integral test to be applied $f\left( x\right) =\dfrac {1}{{2x}+1}$ must be positive continuous and decreasing for $x>1$
We see that function is positive and continuous
$f'\left( x\right) =\dfrac {-2}{\left( 2x+1\right) ^{2}}$
the function is decreasing
So integral test can be applied
$$\int ^{\infty }_{1}\dfrac {1}{2x+1}=\dfrac {1}{2}\ln \left( 2x+1\right) ]^{\infty }_{1}=\infty $$
The integral diverges, so the series diverges