Comparison test for real series/Statement

Grade: D

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Statement

Suppose [ilmath](a_n)_{n\in\mathbb{N} } [/ilmath] and [ilmath](b_n)_{n\in\mathbb{N} } [/ilmath] are real sequences and that we have:

1. [ilmath]\forall n\in\mathbb{N}[a_n\ge 0\wedge b_n\ge 0][/ilmath] - neither sequence is non-negative, and
2. [ilmath]\exists K\in\mathbb{N}\forall n\in\mathbb{N}[n>K\implies a_n\le b_n[/ilmath] - i.e. that eventually [ilmath]a_n\le b_n[/ilmath].

Then:

• if [ilmath]\sum^\infty_{n\eq 1}b_n[/ilmath] converges, so does [ilmath]\sum^\infty_{n\eq 1}a_n[/ilmath]
• if [ilmath]\sum^\infty_{n\eq 1}a_n[/ilmath] diverges so does [ilmath]\sum^\infty_{n\eq 1}b_n[/ilmath]