Chapter 10.5, Problem 37E

a.

To determine

Whether the statement “Suppose $0<a_k<b_k$. If ${\displaystyle \sum a_k}$ converges, then ${\displaystyle \sum b_k}$ converges” is true or false.

b.

To determine

Whether the statement “Suppose $0<a_k<b_k$. If ${\displaystyle \sum a_k}$ diverges, then ${\displaystyle \sum b_k}$ diverges” is true or false.

c.

To determine

Whether the statement “Suppose $0<b_k<c_k<a_k$. If ${\displaystyle \sum a_k}$ converges, then ${\displaystyle \sum b_k}$ and ${\displaystyle \sum c_k}$ converge” is true or false.

d.

To determine

Whether the statement “When applying the Limit Comparison Test, an appropriate comparison series for ${\displaystyle \sum _{k=1}^{\infty }\frac{k^2+2k+1}{k^5+5k+7}}$ is ${\displaystyle \sum _{k=1}^{\infty }\frac{1}{k^3}}$” is true or false.