Plaustrarius

Say I have a power series centered at 7 with radius of convergence 3. Is there any way that this power series could converge at both 4 and 10? Or,stated another way, does a power series exist which has this interval of convergence: [4,10]?If so, could you please give me an example? If not possible, why?

Using the hyperbolic form of Euler's formula ( e^x = coshx + sinhx) I've found that:

arcsinh(tanx) = arcsech(cosx) = ln(secx + tanx)

however, the graphs of these three functions are different, as are their derivatives, because of restrictions that crop up before simplifying, or removable discontinuity.

however, the absolute values for each function and each function's derivative are exactly the same.

thanks in advance! cheers!

What is the area of the largest circle that is inscribed within an equilateral triangle whose perimiter is equal to 3(s)? (basically each side of the triangle is length s)

I'm looking for an answer in general terms, not a quantitative answer.

I got that the area of the circle is (pi(s)^2)/3 using basically only trigonometry, but I have no way of checking that answer really. Thanks in advance!