The logarithm is "entire" for reals > 0 (not quite "entire", but you know what I mean), but it's standard power series diverges outside the range (0, 2]. Analytic extension is used outside that range. So this function might only converge for x<=1.
Besides, you do know what the formula equals at x=5, don't you? Well, for starters, at x=4, it's approximately 5.03481484682034616908489989276E+41. Bear in mind, this function is equal to the second iterated logarithm (base e) of my cheta function, shifted by a constant in the x direction. So it has tetrational growth.
It's like trying to solve sin(x) using the power series, and then saying it appear divergent because you tried to solve the power series for x=100.
So no worries. Just make sure it converges for x=3, which should equal 96.0223655650268799109865292599.
Besides, you do know what the formula equals at x=5, don't you? Well, for starters, at x=4, it's approximately 5.03481484682034616908489989276E+41. Bear in mind, this function is equal to the second iterated logarithm (base e) of my cheta function, shifted by a constant in the x direction. So it has tetrational growth.
It's like trying to solve sin(x) using the power series, and then saying it appear divergent because you tried to solve the power series for x=100.
So no worries. Just make sure it converges for x=3, which should equal 96.0223655650268799109865292599.
~ Jay Daniel Fox