14. Sequences & Series

Determine the convergence or divergence of the series. 

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

Determine the convergence or divergence of the series. 

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

Determine the convergence or divergence of the series. 

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

Determine the convergence or divergence of the series. 

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

What comparison series would you use with the Comparison Test to determine whether

∑ (k = 1 to ∞) 1 / (k² + 1) converges?

What comparison series would you use with the Comparison Test to determine whether ∑ (k = 1 to ∞) 2ᵏ / (3ᵏ + 1) converges?

What comparison series would you use with the Limit Comparison Test to determine whether ∑ (k = 1 to ∞) (k² + k + 5) / (k³ + 3k + 1) converges?

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) 1 / (k² + 4)

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 2 to ∞) (−1)ᵏ (1 + 1/k)

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 1 to ∞) (−1)ᵏ (k¹¹ + 2k⁵ + 1) / [4k(k¹⁰ + 1)]

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k¹/ᵏ

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A series that converges must converge absolutely.