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I am beginner and I want to know some facts about limit as we know them for checking the convergence of a series by ratio test.

If the ratio is greater than $1$ then it is convergent but my point is to know if instead of finding ratio we could find the difference of the $n+1$ and $n^th$ terms then, is there any relation of convergence for that?

Please help.

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    You really should use points and commas. It is very hard to undestand your question. – 2017-01-28
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    @klirk sorry I remember next time – 2017-01-28

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No. For $\sum_{n=1}^{\infty} a_n$ to converge, we must have $a_n \to 0$ so if the series converges we will also have $a_{n+1} - a_n \to 0$. However, if $a_{n+1} - a_n \to 0$ we can't even conclude that $a_n \to 0$ and even if we could, it wouldn't tell us anything about the convergence of the series.