Comparison test for real series/Statement

Grade: D

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Statement

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

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

Then:

• if $\sum^\infty_{n\eq 1}b_n$ converges, so does $\sum^\infty_{n\eq 1}a_n$
• if $\sum^\infty_{n\eq 1}a_n$ diverges so does $\sum^\infty_{n\eq 1}b_n$