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If two sequences converge equally, we have $$\lim_{n\rightarrow \infty }\left ( a_{n} \right )=\lim_{n\rightarrow \infty }\left ( b_{n} \right )$$

As a follow up, is the following equality also true? $$\lim_{n\rightarrow \infty }\left ( \ln a_{n} \right )=\lim_{n\rightarrow \infty }\left ( \ln b_{n} \right )$$

Notice that I didn't put absolute value brackets, because I am working with sequences involving only positive terms at the moment.

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If $\lim_{n\rightarrow \infty }a_{n}=\lim_{n\rightarrow \infty } b_{n} >0$ then you can use that $\ln$ is a continuous function. And, if $\lim_{n\rightarrow \infty }a_{n}=\lim_{n\rightarrow \infty } b_{n} =0$ then you have that $\lim_{n\rightarrow \infty }\ln a_{n}=\lim_{n\rightarrow \infty } \ln b_{n} =-\infty.$

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  • $\begingroup$ Good point there. $\endgroup$ – Shemafied Nov 24 '14 at 19:38
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Yes, because $\mathrm{ln}$ is continuous.

(Of course, the question only makes sense if $a_n>0$ and $b_n>0$ for (almost) all $n$.)

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  • $\begingroup$ Yeah, of course, because even if there were a small amount of negative terms, they still wouldn't play a role in the convergence of a sequence. $\endgroup$ – Shemafied Nov 24 '14 at 19:36

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