Proving Convergence Test: 5 Statements Explained

[rainwyz0706]

There are five statements:
(a) If n^2 an → 0 as n → ∞ then ∑ an converges.
(b) If n an→ 0 as n → ∞ then ∑an converges.
(c) If ∑an converges, then ∑((an )^2)converges.
(d) If ∑ an converges absolutely, then ∑((an )^2) converges.
(e) If ∑an converges absolutely, then |an | < 1/n for all sufficiently large n.

I suppose that a,d,e are true, not quite sure about b,c.
Could anyone please give me some hints how to prove the statements or give some counter-example? Any help is greatly appreciated!

 

[Mark44]

There are five statements:
(a) If n^2 an → 0 as n → ∞ then ∑ an converges.
(b) If n an→ 0 as n → ∞ then ∑an converges.
(c) If ∑an converges, then ∑((an )^2)converges.
(d) If ∑ an converges absolutely, then ∑((an )^2) converges.
(e) If ∑an converges absolutely, then |an | < 1/n for all sufficiently large n.

I suppose that a,d,e are true, not quite sure about b,c.
Could anyone please give me some hints how to prove the statements or give some counter-example? Any help is greatly appreciated!

Start by trying to prove the ones you think are true. For the ones you think are untrue, look at the series whose behavior you know, and see if any might serve as a counterexample.

Show us what you have tried, and we'll take it from there.

 

[Tedjn]

(a) There exists N such that |n²a_n| < 1 for all n > N. Where can you go from there?

(b) Trying the same trick as in (a) doesn't quite work. In fact, a counterexample is the series \sum_{i=2}^\infty \frac{1}{n\log n}. Prove that it is a counterexample.

(c) Think about alternating series.

(d) What is the limit of |a_n| as n tends to infinity? What is the size of a_n² relative to |a_n|?

(e) This is a weird one and precisely as you've stated it, it isn't true. Are you sure you want |a_n| < 1/n where the subscript and denominator are both the same n?

 

[rainwyz0706]

I've got them. Thanks a lot!