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Suppose $\sum _{n=1}^{\infty \:}a_n $ and $\sum _{n=1}^{\infty \:}b_n$ are series with positive terms. Prove that if $\lim_{n\to \infty}(\frac{a_{n}}{b_n} )>0$ then $\sum _{n=1}^{\infty \:}a_n $ converges if and only if $\sum _{n=1}^{\infty \:}b_n$ converges.

I try to use the definition of the limit.

Thank you.

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  • $\begingroup$ This is known as "Limit Comparison Test" whose proof is very well known. $\endgroup$
    – Juniven
    Mar 19 '17 at 16:25
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$\lim\frac{a_n}{b_n}=c>0$ implies that for any $\varepsilon\in(0,c)$ there is some $N_\varepsilon$ ensuring $$ c-\varepsilon \leq \frac{a_n}{b_n} \leq c+\varepsilon $$ for any $n\geq N_\varepsilon$. I leave to you to finish the proof through this lemma.

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