Largest open interval of convergence for the power series about c=0 representing the function f(x)=2/(x^2 +4)?

Largest open interval of convergence for the power series about c=0 representing the function f(x)=2/(x^2 +4)?


...don't worry about endpoints





possible solutions...





(-1/6, 1/6)





(-1/4, 1/4)





(-1/3, 1/3)





(-1/2, 1/2)





(-1, 1)





(-3/2, 3/2)





(-2, 2)





(-3, 3)





(-4, 4)





or is it none of these ?

Largest open interval of convergence for the power series about c=0 representing the function f(x)=2/(x^2 +4)?
Need to manipulate this into an infinite series by recognizing geometric series converges to S = a / (1-r) for IrI %26lt; 1





2/ (x^2+4)


2/ (1 - (-x^2-3)


therefore a=2


and r = (-x^2 - 3)


need IrI%26lt;1


I-x^2-3I %26lt; 1


which has no solutions


Largest interval is only c=0
Reply:(-2, 2)





The only poles are at x^2 + 4 = 0 or x = +/- 2i, so in the complex plane the series converges for all z with |z| %26lt; 2. For just the reals that works out to (-2, 2).





Dan