Chapter 10.4, Problem 47E

a.

To determine

Whether the statement “The sum ${\displaystyle \sum _{k=1}^{\infty }\frac{1}{3^k}}$ is a p-series” is true or false.

b.

To determine

Whether the statement “The sum ${\displaystyle \sum _{k=3}^{\infty }\frac{1}{\sqrt{k-2}}}$ is a p-series” is true or false.

c.

To determine

Whether the given statement is true or not and give an explanation or counter example.

d.

To determine

Whether the given statement is true or not and give an explanation or counter example.

e.

To determine

Whether the statement “If ${\displaystyle \sum k^{-p}}$ converges, then ${\displaystyle \sum k^{-p+0.001}}$ converges” is true or not and give an explanation or counter example.

f.

To determine

Whether the statement “If $\underset{k\to \infty }{\lim }{a}_k=0$, then ${\displaystyle \sum a_k}$ converges” is true or not and give an explanation or counter example.