Question

Suppose an and bn are series with positive terms and bn is known to be convergent. (a) If an > bn for all n, what can you say about an ? Why? an converges by the Comparison Test. We cannot say anything about an . an diverges by the Comparison Test. an converges if and only if an ≤ 4bn. an converges if and only if an ≤ 2bn. (b) If an < bn for all n, what can you say about an ? Why? an diverges by the Comparison Test. an converges if and only if bn 2 ≤ an ≤ bn. We cannot say anything about an . an converges by the Comparison Test. an converges if and only if bn 4 ≤ an ≤ bn.

Answer 1

Answer:

a. i. We cannot say anything about aₙ

ii. we do not know if the terms of aₙ are increasing or decreasing

b. i. aₙ converges by the Comparison Test.

ii. aₙ also converges since bₙ.

Step-by-step explanation:

(a)

i) If aₙ > bₙ for all n, what can you say about aₙ ? Why?

We cannot say anything about an.

ii) Why?

This is because since ∑bₙ converges, ∑aₙ > ∑bₙ, we do not know if the terms of aₙ are increasing or decreasing, so we have no information about it.  

(b) If aₙ < bₙ for all n, what can you say about aₙ ? Why?

i) aₙ converges by the Comparison Test.

ii) Why?

This is because since ∑bₙ converges, and ∑aₙ < ∑bₙ, we know that the terms of aₙ are also decreasing and, by the comparison test, aₙ also converges since bₙ.