Without using a calculator, how would you determine if terms of the form are positive? I am mainly interested in the case where are integers.
When there are 5 or fewer terms involved, we can try and split the terms and square both sides, to reduce the number of surds that are involved. For example, to determine if we can square both sides of to obtain
Repeated squaring eventually resolves this question, as the number of surds are reduced. This is a fail-proof method, regardless of the numbers that are involved.
However, when there are more than 6 terms involved, then repeated squaring need not necessarily reduce the terms that are involved.
E.g. How would you determine if
I can think of several approaches
There are special cases, which allow us to apply Jensen's inequality. However, this gives a somewhat restrictive condition on the set of values.
Show that However, it might not be feasible to guess what the middle number is, unless you already had a calculator. Taylor approximations might offer an approximation, though the bound could be very tight (and hence require many terms).
Do you have any other suggestions?