14. Sequences & Series

Determine whether the given series are convergent using the Ratio Test.

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

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

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

Use the Alternating Series Test to determine if the following series is conditionally convergent, absolutely convergent, or divergent.

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

Determine whether the given series is convergent.

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

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

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

Determine if the following series converges, diverges, or is inconclusive. 

${\sum }_{n=1}^{\infty }{\left(\frac{2n^2-1}{n^2+5}\right)}^{-n}$

Determine whether the given series is convergent.

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

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) ${\sum }_{n=1}^{\infty }\frac{1}{3n+2}$

Determine the convergence or divergence of the series. 

${\sum }_{k=1}^{\infty }\frac{7k^2}{5k+3}$

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

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

Determine whether the given series is convergent using the Ratio Test.

${\sum }_{n=1}^{\infty }\frac{n^5}{{\left(6\right)}^n}$

Use the Alternating Series Test to approximate the sum of the series using the first five terms. 

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

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

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

Use the Direct Comparison Test to determine whether the series converges.

${\sum }_{n=1}^{\infty }\frac{3}{\sqrt{n}-2}$

Use the Limit Comparison Test to determine whether the series converges.

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