Solving an difficult inequality

[pinodk]

Ill take it from the top please excuse me for any misspellings.

In Numerical analysis, I have to show that a function only converges towards a solution in a certain interval.
The function is

f(x)=x/8+arctg(x)

Im using Newtons method, i.e.

xk+1 = xk - f(x)/f'(x)

And i know it will converge if I am in an interval that satisfies

|xk-xk+1| < 2xk

I then use that

f(x)/f'(x) < 2xk

giving me

1+8*arctg(x)/x
--------------- < 2
1+8/1+x^2

My problem is now how to solve this inequality, I'm trying with mathematica, but i can't make it work
Its not important how I solve it, i just need a solution, and guidelines to finding it...
Anyone up for the task?

 

[Fermat]

I rearranged your inequality to get,

x < tan{x(x²+17)/8(x²+1)}

I plotted y = tan{x(x²+17)/8(x²+1)} and y = x using graphmatica and I got y = x < y = tan{x(x²+17)/8(x²+1)} for x < 1.829 (approx.)

 

[pinodk]

Thanks!
Sometimes I hate myself :-(
The problem for mathematica is obviously the arctg expression, so y the h*** didnt i think of reaaranging it...
i will go do that now... thanks man!