14. Sequences & Series

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

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

Determine the convergence or divergence of the series. 

${\sum }_{n=1}^{\infty }\frac{5^{n-1}}{2^n}$

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

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

Determine whether the given series is convergent.

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

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

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

Determine whether the given series are convergent.

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

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

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

Determine the convergence or divergence of the series. 

${\sum }_{n=1}^{\infty }\frac{n\bullet 8^n}{(n+1)!}$

Use the Alternating Series Test to determine the convergence or divergence of the series.

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

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.

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

Determine the convergence or divergence of the series. 

${\sum }_{k=1}^{\infty }\frac{{(-1)}^{k}4}{3k+2}$

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

${\sum }_{k=1}^{\infty }\frac{\sin \frac{n\pi }{2}}{n^3}$

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

${\sum }_{n=1}^{\infty }{(-1)}^{n-1}\left(\frac{3}{4n+5}\right)$

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

${\sum }_{n=1}^{\infty }{\left(\frac{4n+1}{n-2}\right)}^n$