A general convergent series under different circumstances.

[uber_kim]

Homework Statement

Ʃa_n (sum from n=1 to ∞) converges.

1) Determine whether the series Ʃln(1+a_n) (sum from n=1 to ∞) converges or diverges. Assume that a_n>0 for all n.

2) Show each of the following statements or give a counter-example that establishes that it is false:
a)Ʃa_i² (sum i=1 to ∞) converges if the a_i are alternating.
b)Ʃa_i² converges if the a_i are non-negative.

Homework Equations

The Attempt at a Solution

For the first question (1), if a_n converges, then the terms go to zero as n goes to ∞. This means that as n goes to ∞ for Ʃln(1+a_n), the series will diverge to -∞, because ln(0)=-∞. I'm not sure if that's right, and if that's a good enough proof. Any ideas?

For the second question (2a), if Ʃa_i converges, then the square should converge, but faster. I'm not sure how it alternating would effect the convergence, since the absolute value of the terms are still getting smaller as i gets bigger.

(2b) The same as the last one, I don't see how the terms being positive makes a difference, since the series is still converging. It just means that the terms don't dip below zero.

Not sure if I'm on the right track or not. Any thoughts would be greatly appreciated.

Thanks!

 

[jbunniii]

For the first question (1), if a_n converges, then the terms go to zero as n goes to ∞. This means that as n goes to ∞ for Ʃln(1+a_n), the series will diverge to -∞, because ln(0)=-∞. I'm not sure if that's right, and if that's a good enough proof. Any ideas?

This isn't right. If a_n goes to 0, then 1 + a_n goes to 1, so \ln(1 + a_n) goes to zero. Thus you have not ruled out convergence.

For the second question (2a), if Ʃa_i converges, then the square should converge, but faster. I'm not sure how it alternating would effect the convergence, since the absolute value of the terms are still getting smaller as i gets bigger.

Consider an alternating series that converges, but not absolutely.

(2b) The same as the last one, I don't see how the terms being positive makes a difference, since the series is still converging. It just means that the terms don't dip below zero.

Hint: How does a_n^2 compare with a_n if 0 \leq a_n \leq 1?