Let a, b be two positive real numbers. Show that arctan((a+1)/b) - arctan(a/b) < b/(a^2 + b^2)
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This looks like a mean value theorem question.
It's too late for me to try it now, i'll have a go in the morning.
Originally Posted by kjwill1776 Let a, b be two positive real numbers. Show that The first thing to note is that is a decreasing function. Hence if
Now use the Mean Value Theorem: for
From above though, we know that implies
The rest is below if you need it. Spoiler: Thus,
Multiplying both sides by gives us:
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