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I was reading Binomial theorem of any index of Higher Algebra by Hall & Knight; there a section was attributed to find the greatest term in the expansion of $(1+x)^n$ for any rational value of $n$. The authors caution that for having the greatest term for the expansion when $n$ being a positive fraction or negative, $x$ can't be greater than unity. They reasoned as the following line goes:

If $x$ be greater than unity, by increasing $r$, the above multiplier can be made as near as we please to $-x$; so that after a certain term each term is nearly $x$ times the preceding term numerically, & thus the terms increase continually, & there is no greatest term.

I couldn't understand their reasoning. What does really happen after we come to the so-called largest term as $r = \text{integral part of} \dfrac{(n+1)x}{1+x} $ when $x>1$? Can anyone please explain what the authors want to tell & also why the greatest term exists for $x<1$??

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