Chapter 8.4, Problem 51E

a.

To determine

The statement “If ${\displaystyle \sum _{k=1}^{\infty }{a}_k}$ converges, then ${\displaystyle \sum _{k=10}^{\infty }{a}_k}$ converges” is true or not and give an explanation or counter example.

b.

To determine

The statement “If ${\displaystyle \sum _{k=1}^{\infty }{a}_k}$ diverges, then ${\displaystyle \sum _{k=10}^{\infty }{a}_k}$ diverges” is true or not and give an explanation or counter example.

c.

To determine

The statement “If ${\displaystyle \sum a_k}$ converges, then ${\displaystyle \sum \left(a_k+0.0001\right)}$ also converges” is true or not and give an explanation or counter example.

d.

To determine

The statement “If ${\displaystyle \sum p^k}$ diverges, then ${\displaystyle \sum {\left(p+0.001\right)}^k}$ diverges, for a fixed real number p” is true or not and give an explanation or counter example.

e.

To determine

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

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