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Why is the series $a_n={1\over2^n+n}$ convergent whereas the series $a_n={3^n+1\over4^n+5}$ is absolutely convergent? Since both use the comparison test to the geometric series, what makes one abs. convergent and the other not? I know a series converges absolutely if $|a_n|$ converges so why the difference?

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A series that is positive is absolutely convergent as soon as it is convergent. So both are absolutely convergent.

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  • $\begingroup$ Thank you. That's what I thought.. $\endgroup$ – user71865 Apr 10 '13 at 15:51
  • $\begingroup$ (That's what I thought) vs (what makes one abs. convergent and the other not?) $\endgroup$ – Did Apr 10 '13 at 16:00

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