Question

Which of the following statements are always true and which are sometimes false? If the answer is sometimes false, give a counterexample; that is, an example of a sequence which demonstrates that the statement cannot always be true. If the answer is always true, indicate why by referencing a theorem or an idea in the book.

Answer 1

Answer:

Explanation:

1.) If {an} > 0 for all n and limit approaches n of an = L, then L > 0 ALWAYS TRUE

2.) If {an} is bounded, then it converges

can be FALSE ; a_n = (-1)^n does not converge

3.) If {an} is decreasing, then it converges

can be FALSE ; a_n = -n does not converge

4.) If {an} is decreasing and {an} > 0 for all n, then it converges

TRUE

5.) If {an} is bounded, then {an/n} converges to 0

TRUE

6.) If [an} converges and {bn} converges then {an/bn} converges

can be FALSE; a_n = 1/n and b_n = (-1)^n/n

Both converge to 0 but a_n/b_n = (-1)^n which does not converge