Hi everyone. In a proof I'm working on, I have a ratio of two sums of functions in the following form:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\frac{f_1(x)+f_2(x)+...f_n(x)}{g_1(x)+g_2(x)+...+g_n(x)}[/itex]

I want to prove this ratio is monotonically increasing in [itex]x[/itex]. All of the functions [itex]f_i(x)[/itex] and [itex]g_i(x)[/itex] are positive and also (importantly) I know that for all [itex]i=1,2,...,n[/itex], the ratio [itex]f_i(x)/g_i(x)[/itex] is monotonically increasing in [itex]x[/itex], i.e. [itex]f_1(x)/g_1(x)[/itex] is increasing in [itex]x[/itex], [itex]f_2(x)/g_2(x)[/itex] is increasing in [itex]x[/itex], etc.

Is there a simple way to prove this without requiring further information about these functions? I've been stuck on it for a while. Does it have to be true that the ratio of the sums is increasing? If anyone can suggest a straightforward approach (or tell me if it's not possible without further information) I'd be very grateful, thanks!

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# Proving monotonicity of a ratio of two sums

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