I am trying to understand something in the proof of the ratio test for series convergence.(adsbygoogle = window.adsbygoogle || []).push({});

If [itex]a_{n}[/itex] is a sequence of positive numbers, and that the ratio test shows that [itex] \lim_{n→∞}\frac{a_{n+1}}{a_{n}} = r < 1[/itex], then the series converges.

Apparently, the proof defines a number R : r<R<1, and then shows that there exists a N>0 such that [itex]\frac{a_{n+1}}{a_{n}} < R [/itex] for all n>N.It need not to be true in the case where n=N, right?Up to this part I get.

But then it concludes from the above that, there exists a positive N such that

[itex] a_{N+1}<a_{n}R [/itex] which does not follow due to the statement in bold.

Could someone please point out where I am wrong so I can continue this theorem without any qualms? Thanks!

BiP

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# Proof of the ratio test

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