Use the divergence test to determine if the following series diverge or state that the test is inconclusive. 
${\sum }_{n=1}^{\infty }\sin \left(\frac{n\pi }{2}\right)$

If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.

In Exercises 67–72, use the results of Exercises 63 and 64 to determine if each series converges or diverges.

∑(from n=2 to ∞) [(ln n)¹⁰⁰⁰ / n¹.⁰⁰¹]

Determining Convergence of Series

Which of the series in Exercises 25–44 converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers.

∑ (from n = 2 to ∞) 1/[n(ln n)²]

Use the divergence test to determine if the following series diverge or state that the test is inconclusive. 

${\sum }_{n=1}^{\infty }\frac{{10}^n}{n!}$

Use the divergence test to determine if the following series diverge or state that the test is inconclusive. 

${\sum }_{n=1}^{\infty }\frac{n^2}{n(n^2-1000)}$

Explain why the integral test does not apply to the series.

${\sum }_{n=1}^{\infty }\frac{{(-1)}^n}{\ln n}$

Explain why the integral test does not apply to the series.

${\sum }_{n=4}^{\infty }\frac{1}{n+\sin n}$