14. Sequences & Series

Confirm that the integral test applies and then use the integral test to determine convergence of the series.

(A) ${\sum }_{n=1}^{\infty }\frac{arc\tan n}{n^2+1}$ (B) ${\sum }_{n=1}^{\infty }\frac{1}{3n+2}$

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) k / eᵏ

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.

∑ (from k = 1 to ∞)ln((2k + 1) / (2k − 1))

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 1 to ∞) 1 / k^(1 + p),p > 0

Use the Limit Comparison Test to determine if the following series converges.

${\sum }_{n=1}^{\infty }\frac{n^2+1}{n}$

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

${\sum }_{n=1}^{\infty }\sin \left(\frac{n\pi }{2}\right)$

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.

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

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

${\sum }_{n=0}^{\infty }\frac{n}{n^2-1}$

Determine whether the given series is convergent.

${\sum }_{n=1}^{\infty }\frac{1}{{}^{{}^5}\sqrt{n^3}}$

Determine whether the given series is convergent.

${\sum }_{n=1}^{\infty }\frac{5}{3n-7}$

Determine whether the given series are convergent.

${\sum }_{n=1}^{\infty }\frac{n^e}{n^2}$

Determine whether the given series is convergent.

${\sum }_{n=1}^{\infty }{n}^{-1}+n^{-3}$