14. Sequences & Series

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.

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

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

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

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

${\sum }_{n=1}^{\infty }\frac{{\cos }^{2}n}{n^2}$ Hint: Compare to $b_n=\frac{1}{n^2}$

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

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

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

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