Question

Let \(\left<x_n\right>\) and \(\left<y_n\right>\) be real sequences and let for some k ϵ N, 0 ≤ x_n ≤ y_n for n ≥ k.

Then, which of the following statements is true?

1. Divergence of ∑ y_n ⇒ Divergence of ∑ x_n
2. ∑ x_n and ∑ y_n are always divergent
3. Convergence of ∑ x_n ⇒ Convergence of ∑ y_n
4. Convergence of ∑ y_n ⇒ Convergence of ∑ x_n

Answer 1

Correct Answer - Option 4 : Convergence of ∑ y_n ⇒ Convergence of ∑ x_n

Concept:

The convergence or divergence of sequences is tested by various methods. One is Comparison test.

Comparison test:

Suppose that we have two series ∑a_n and ∑b_n with a_n, b_n ≥ 0 for all n and a_n ≤ b_n for all n. Then, 

1. If ∑b_n is convergent then so is ∑a_n

2. If ∑a_n is divergent then so is ∑b_n

The statement 1 is given in the 4th option.