Determine whether the series ∑ n=1
[infinity]
​
2n
ln(2n)
​
converges or diverges. SOLUTION The function f(x)= (2x)
ln(2x)
​
is positive and continuous for x>□ because the logarith so we compute its derivative: f ′
(x)= 4x 2
(1)2x−2ln(2x)
​
= 2x 2
Thus f ′
(x)⟨0 when ln(2x)⟩, that is, x⟩. It follows that f is decreasing whe ∫ 1
[infinity]
​
2x
ln(2x)
​
dx=lim t→[infinity]
​
∫ 1
t
​
2x
ln(2x)
​
dx =lim t→[infinity]
​
4
(ln(2t)) 2
​
− 4
(ln(2)) 2
​
=[infinity] Since this improper integral is divergent, the series ∑ n=1
[infinity]
​
2n
ln(2n)
​
is also divergent by the Integral Test.